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Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 1 - Differentiation [Latest edition]

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Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board - Shaalaa.com
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Chapter 1: Differentiation

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Miscellaneous Exercise 1
Exercise 1.1 [Pages 11 - 13]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Differentiation Exercise 1.1 [Pages 11 - 13]

Exercise 1.1 | Q 1.1 | Page 11

Differentiate the following w.r.t.x: (x3 – 2x – 1)5 

Exercise 1.1 | Q 1.2 | Page 11

Differentiate the following w.r.t.x: `(2x^(3/2) - 3x^(4/3) - 5)^(5/2)`

Exercise 1.1 | Q 1.3 | Page 11

Differentiate the following w.r.t.x: `sqrt(x^2 + 4x - 7)`

Exercise 1.1 | Q 1.4 | Page 11

Differentiate the following w.r.t.x: `sqrt(x^2 + sqrt(x^2 + 1)`

Exercise 1.1 | Q 1.5 | Page 11

Differentiate the following w.r.t.x: `(3)/(5root(3)((2x^2 - 7x - 5)^5`

Exercise 1.1 | Q 1.6 | Page 11

Differentiate the following w.r.t.x: `(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`

Exercise 1.1 | Q 2.01 | Page 12

Differentiate the following w.r.t.x: cos(x2 + a2)

Exercise 1.1 | Q 2.02 | Page 12

Differentiate the following w.r.t.x: `sqrt(e^((3x + 2) +  5)`

Exercise 1.1 | Q 2.03 | Page 12

Differentiate the following w.r.t.x: `log[tan(x/2)]`

Exercise 1.1 | Q 2.04 | Page 12

Differentiate the following w.r.t.x: `sqrt(tansqrt(x)`

Exercise 1.1 | Q 2.05 | Page 12

Differentiate the following w.r.t.x: cot3[log(x3)]

Exercise 1.1 | Q 2.06 | Page 12

Differentiate the following w.r.t.x: `5^(sin^3x + 3)`

Exercise 1.1 | Q 2.07 | Page 12

Differentiate the following w.r.t.x: `"cosec"(sqrt(cos x))`

Exercise 1.1 | Q 2.08 | Page 12

Differentiate the following w.r.t.x: log[cos(x3 – 5)]

Exercise 1.1 | Q 2.09 | Page 12

Differentiate the following w.r.t.x: `e^(3sin^2x - 2cos^2x)`

Exercise 1.1 | Q 2.1 | Page 12

Differentiate the following w.r.t.x: cos2[log(x2 + 7)]

Exercise 1.1 | Q 2.11 | Page 12

Differentiate the following w.r.t.x: tan[cos (sinx)]

Exercise 1.1 | Q 2.12 | Page 12

Differentiate the following w.r.t.x: sec[tan (x4 + 4)]

Exercise 1.1 | Q 2.13 | Page 12

Differentiate the following w.r.t.x: `e^(log[(logx)^2 - logx^2]`

Exercise 1.1 | Q 2.14 | Page 12

Differentiate the following w.r.t.x: `sinsqrt(sinsqrt(x)`

Exercise 1.1 | Q 2.15 | Page 12

Differentiate the following w.r.t.x: `log[sec (e^(x^2))]`

Exercise 1.1 | Q 2.16 | Page 12

Differentiate the following w.r.t.x: loge2(logx)

Exercise 1.1 | Q 2.17 | Page 12

Differentiate the following w.r.t.x: [log {log(logx)}]2

Exercise 1.1 | Q 2.18 | Page 12

Differentiate the following w.r.t.x: sin2x2 – cos2x2 

Exercise 1.1 | Q 3.01 | Page 12

Differentiate the following w.r.t.x: (x2 + 4x + 1)3 + (x3− 5x − 2)4 

Exercise 1.1 | Q 3.02 | Page 12

Differentiate the following w.r.t.x: (1 + 4x)5 (3 + x −x2)

Exercise 1.1 | Q 3.03 | Page 12

Differentiate the following w.r.t.x: `x/(sqrt(7 - 3x)`

Exercise 1.1 | Q 3.04 | Page 12

Differentiate the following w.r.t.x: `(x^3 - 5)^5/(x^3 + 3)^3`

Exercise 1.1 | Q 3.05 | Page 12

Differentiate the following w.r.t.x: (1 + sin2 x)2 (1 + cos2 x)3 

Exercise 1.1 | Q 3.06 | Page 12

Differentiate the following w.r.t.x: `sqrt(cosx) + sqrt(cossqrt(x)`

Exercise 1.1 | Q 3.07 | Page 12

Differentiate the following w.r.t.x: log (sec 3x+ tan 3x)

Exercise 1.1 | Q 3.08 | Page 12

Differentiate the following w.r.t.x: `(1 + sinx°)/(1 - sinx°)`

Exercise 1.1 | Q 3.09 | Page 12

Differentiate the following w.r.t.x: `cot(logx/2) - log(cotx/2)`

Exercise 1.1 | Q 3.1 | Page 12

Differentiate the following w.r.t.x: `(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`

Exercise 1.1 | Q 3.11 | Page 12

Differentiate the following w.r.t.x: `(e^sqrt(x) + 1)/(e^sqrt(x) - 1)`

Exercise 1.1 | Q 3.12 | Page 12

Differentiate the following w.r.t.x: log[tan3x.sin4x.(x2 + 7)7]

Exercise 1.1 | Q 3.13 | Page 12

Differentiate the following w.r.t.x: `log(sqrt((1 - cos3x)/(1 + cos3x)))`

Exercise 1.1 | Q 3.14 | Page 12

Differentiate the following w.r.t.x: `(sqrt((1 + cos((5x)/2))/(1 - cos((5x)/2))))`

Exercise 1.1 | Q 3.15 | Page 12

Differentiate the following w.r.t.x: `log(sqrt((1 - sinx)/(1 + sinx)))`

Exercise 1.1 | Q 3.16 | Page 12

Differentiate the following w.r.t.x: `log[4^(2x)((x^2 + 5)/(sqrt(2x^3 - 4)))^(3/2)]`

Exercise 1.1 | Q 3.17 | Page 12

Differentiate the following w.r.t.x: `log[(ex^2(5 - 4x)^(3/2))/root(3)(7 - 6x)]`

Exercise 1.1 | Q 3.18 | Page 12

Differentiate the following w.r.t.x:

`log[a^(cosx)/((x^2 - 3)^3 logx)]`

Exercise 1.1 | Q 3.19 | Page 12

Differentiate the following w.r.t.x: y = `(25)^(log_5(secx)) - (16)^(log_4(tanx)`

Exercise 1.1 | Q 3.2 | Page 12

Differentiate the following w.r.t.x: `(x^2 + 2)^4/(sqrt(x^2 + 5)`

Exercise 1.1 | Q 4.1 | Page 12

A table of values of f, g, f' and g' is given :

x f(x) g(x) f'(x) g'(x)
2 1 6 –3 4
4 3 4 5 -6
6 5 2 –4 7

If r(x) =f [g(x)] find r' (2).

Exercise 1.1 | Q 4.2 | Page 12

A table of values of f, g, f' and g' is given :

x f(x) g(x) f'(x) g'(x)
2 1 6 –3 4
4 3 4 5 -6
6 5 2 –4 7

If R(x) =g[3 + f(x)] find R'(4).

Exercise 1.1 | Q 4.3 | Page 12

A table of values of f, g, f' and g' is given :

x f(x) g(x) f'(x) g'(x)
2 1 6 –3 4
4 3 4 5 -6
6 5 2 –4 7

If s(x) =f[9 − f (x)] find s'(4).

Exercise 1.1 | Q 4.4 | Page 12

A table of values of f, g, f' and g' is given :

x f(x) g(x) f'(x) g'(x)
2 1 6 –3 4
4 3 4 5 -6
6 5 2 –4 7

If S(x) =g [g(x)] find S'(6).

Exercise 1.1 | Q 5 | Page 12

Assume that `f'(3) = -1,"g"'(2) = 5, "g"(2) = 3 and y = f["g"(x)], "then" ["dy"/"dx"]_(x = 2) = ?`

Exercise 1.1 | Q 6 | Page 12

If h(x) = `sqrt(4f(x) + 3"g"(x)), f(1) = 4, "g"(1) = 3, f'(1) = 3, "g"'(1) = 4, "find h"'(1)`.

Exercise 1.1 | Q 7 | Page 12

Find the x co-ordinates of all the points on the curve y = sin 2x − 2 sin x, 0 ≤ x < 2π, where `"dy"/"dx"` = 0.

Exercise 1.1 | Q 8 | Page 13

Select the appropriate hint from the hint basket and fill up the blank spaces in the following paragraph. [Activity]:

"Let f (x) =x2 + 5 and g (x) =ex + 3 then
f[g(x)] = .......... and g[f(x)] =...........
Now f'(x) = .......... and g'(x) = ..........
The derivative of f[g(x)] w. r. t. x in terms of f and g is ..........

Therefore `"d"/"dx"[f["g"(x)]]` = .......... and

`["d"/"dx"[f["g"(x)]]]_(x  =  0)` = ..........
The derivative of g[f(x)] w. r. t. x in terms of f and g is

Therefore `"d"/"dx"["g"[f(x)]]` = .......... and

`["d"/"dx"["g"[f(x)]]]_(x  = -1)` = .........."

Hint basket : `{f'["g"(x)]·"g"'(x), 2e^(2x) + 6e^x, 8, "g"' [ f (x)]· f'(x),2xe^(x^2+5),  − 2e^6,e^(2x) + 6e^x + 14, e^(x^2+5) + 3, 2x, e^x}`

Exercise 1.2 [Pages 29 - 30]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Differentiation Exercise 1.2 [Pages 29 - 30]

Exercise 1.2 | Q 1.1 | Page 29

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f-1(y) in the following: y = `sqrt(x)`

Exercise 1.2 | Q 1.2 | Page 29

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f-1(y) in the following: y = `sqrt(2 - sqrt(x)`

Exercise 1.2 | Q 1.3 | Page 29

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f-1(y) in the following: y = `root(3)(x - 2)`

Exercise 1.2 | Q 1.4 | Page 29

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = log(2x – 1)

Exercise 1.2 | Q 1.5 | Page 29

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = 2x + 3

Exercise 1.2 | Q 1.6 | Page 29

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = ex – 3

Exercise 1.2 | Q 1.7 | Page 29

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = e2x-3 

Exercise 1.2 | Q 1.8 | Page 29

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = `log_2(x/2)`

Exercise 1.2 | Q 2.1 | Page 29

Find the derivative of the inverse function of the following : y = x2·ex 

Exercise 1.2 | Q 2.2 | Page 29

Find the derivative of the inverse function of the following : y = x cos x

Exercise 1.2 | Q 2.3 | Page 29

Find the derivative of the inverse function of the following : y = x ·7

Exercise 1.2 | Q 2.4 | Page 29

Find the derivative of the inverse function of the following : y = x2 + log x

Exercise 1.2 | Q 2.5 | Page 29

Find the derivative of the inverse function of the following : y = x log x

Exercise 1.2 | Q 3.1 | Page 29

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = x5 + 2x3 + 3x, at x = 1

Exercise 1.2 | Q 3.2 | Page 29

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = ex + 3x + 2

Exercise 1.2 | Q 3.3 | Page 29

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = 3x2 + 2logx3 

Exercise 1.2 | Q 3.4 | Page 29

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = sin(x – 2) + x2 

Exercise 1.2 | Q 4 | Page 29

If f(x) = x3 + x – 2, find (f–1)'(–2).

Exercise 1.2 | Q 5.1 | Page 29

Using derivative, prove that: tan –1x + cot–1x = `pi/(2)`

Exercise 1.2 | Q 5.2 | Page 29

Using derivative, prove that: sec–1x + cosec–1x = `pi/(2)`    ...[for |x| ≥ 1]

Exercise 1.2 | Q 6.01 | Page 29

Differentiate the following w.r.t. x : tan–1(log x)

Exercise 1.2 | Q 6.02 | Page 29

Differentiate the following w.r.t. x : cosec–1 (e–x)

Exercise 1.2 | Q 6.03 | Page 29

Differentiate the following w.r.t. x : cot–1(x3)

Exercise 1.2 | Q 6.04 | Page 29

Differentiate the following w.r.t. x : cot–1(4x)

Exercise 1.2 | Q 6.05 | Page 29

Differentiate the following w.r.t. x : `tan^-1(sqrt(x))`

Exercise 1.2 | Q 6.06 | Page 29

Differentiate the following w.r.t. x : `sin^-1(sqrt((1 + x^2)/2))`

Exercise 1.2 | Q 6.07 | Page 29

Differentiate the following w.r.t. x : cos–1(1 –x2)

Exercise 1.2 | Q 6.08 | Page 29

Differentiate the following w.r.t. x : `sin^-1(x^(3/2))`

Exercise 1.2 | Q 6.09 | Page 29

Differentiate the following w.r.t. x : cos3[cos–1(x3)]

Exercise 1.2 | Q 6.1 | Page 29

Differentiate the following w.r.t. x : `sin^4[sin^-1(sqrt(x))]`

Exercise 1.2 | Q 7.01 | Page 29

Differentiate the following w.r.t. x : `cot^-1[cot(e^(x^2))]`

Exercise 1.2 | Q 7.02 | Page 29

Differentiate the following w.r.t. x : `"cosec"^-1[1/cos(5^x)]`

Exercise 1.2 | Q 7.03 | Page 29

Differentiate the following w.r.t. x : `cos^-1(sqrt((1 + cosx)/2))`

Exercise 1.2 | Q 7.04 | Page 29

Differentiate the following w.r.t. x : `cos^-1(sqrt(1 - cos(x^2))/2)`

Exercise 1.2 | Q 7.05 | Page 29

Differentiate the following w.r.t. x : `tan^-1[(1 - tan(x/2))/(1 + tan(x/2))]`

Exercise 1.2 | Q 7.06 | Page 29

Differentiate the following w.r.t. x : `"cosec"^-1((1)/(4cos^3 2x - 3cos2x))`

Exercise 1.2 | Q 7.07 | Page 29

Differentiate the following w.r.t. x : `tan^-1[(1 + cos(x/3))/(sin(x/3))]`

Exercise 1.2 | Q 7.08 | Page 29

Differentiate the following w.r.t. x : `cot^-1((sin3x)/(1 + cos3x))`

Exercise 1.2 | Q 7.09 | Page 30

Differentiate the following w.r.t. x : `tan^-1((cos7x)/(1 + sin7x))`

Exercise 1.2 | Q 7.1 | Page 30

Differentiate the following w.r.t. x : `tan^-1(sqrt((1 + cosx)/(1 - cosx)))`

Exercise 1.2 | Q 7.11 | Page 30

Differentiate the following w.r.t. x : tan–1 (cosec x + cot x)

Exercise 1.2 | Q 7.12 | Page 30

Differentiate the following w.r.t. x : `cot^-1[(sqrt(1 + sin  ((4)/3)) + sqrt(1 - sin  ((4x)/3)))/(sqrt(1 + sin  ((4x)/3)) - sqrt(1 - sin  ((4x)/3)))]`

Exercise 1.2 | Q 8.1 | Page 30

Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`

Exercise 1.2 | Q 8.2 | Page 30

Differentiate the following w.r.t. x : `cos^-1((sqrt(3)cosx - sinx)/(2))`

Exercise 1.2 | Q 8.3 | Page 30

Differentiate the following w.r.t. x : `sin^-1((cossqrt(x) + sinsqrt(x))/sqrt(2))`

Exercise 1.2 | Q 8.4 | Page 30

Differentiate the following w.r.t. x : `cos^-1((3cos3x - 4sin3x)/5)`

Exercise 1.2 | Q 8.5 | Page 30

Differentiate the following w.r.t. x : `cos^-1[(3cos(e^x) + 2sin(e^x))/sqrt(13)]`

Exercise 1.2 | Q 8.6 | Page 30

Differentiate the following w.r.t. x : `"cosec"^-1[(10)/(6sin(2^x) - 8cos(2^x))]`

Exercise 1.2 | Q 9.01 | Page 30

Differentiate the following w.r.t. x : `cos^-1((1 - x^2)/(1 + x^2))`

Exercise 1.2 | Q 9.02 | Page 30

Differentiate the following w.r.t. x : `tan^-1((2x)/(1 - x^2))`

Exercise 1.2 | Q 9.03 | Page 30

Differentiate the following w.r.t. x : `sin^-1((1 - x^2)/(1 + x^2))`

Exercise 1.2 | Q 9.04 | Page 30

Differentiate the following w.r.t. x : `sin^-1(2xsqrt(1 - x^2))`

Exercise 1.2 | Q 9.05 | Page 30

Differentiate the following w.r.t. x : cos–1(3x – 4x3)

Exercise 1.2 | Q 9.06 | Page 30

Differentiate the following w.r.t. x : `cos^-1((e^x -  e^(-x))/(e^x +  e^(-x)))`

Exercise 1.2 | Q 9.07 | Page 30

Differentiate the following w.r.t. x : `cos^-1  ((1 - 9^x))/((1 + 9^x)`

Exercise 1.2 | Q 9.08 | Page 30

Differentiate the following w.r.t. x : `sin^-1(4^(x + 1/2)/(1 - 2^(4x)))`

Exercise 1.2 | Q 9.09 | Page 30

Differentiate the following w.r.t. x : `sin^-1  ((1 - 25x^2)/(1 + 25x^2))`

Exercise 1.2 | Q 9.1 | Page 30

Differentiate the following w.r.t. x : `sin^-1  ((1 - x^3)/(1 + x^3))`

Exercise 1.2 | Q 9.11 | Page 30

Differentiate the following w.r.t. x : `tan^-1((2x^(5/2))/(1 - x^5))`

Exercise 1.2 | Q 9.12 | Page 30

Differentiate the following w.r.t. x : `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`

Exercise 1.2 | Q 10.1 | Page 30

Differentiate the following w.r.t. x : `tan^-1((8x)/(1 - 15x^2))`

Exercise 1.2 | Q 10.2 | Page 30

Differentiate the following w.r.t. x : `cot^-1((1 + 35x^2)/(2x))`

Exercise 1.2 | Q 10.3 | Page 30

Differentiate the following w.r.t. x : `tan^-1((2sqrt(x))/(1 + 3x))`

Exercise 1.2 | Q 10.4 | Page 30

Differentiate the following w.r.t. x : `tan^-1[(2^x + 2)/(1 - 3(4^x))]`

Exercise 1.2 | Q 10.5 | Page 30

Differentiate the following w.r.t. x : `tan^-1((2^x)/(1 + 2^(2x + 1)))`

Exercise 1.2 | Q 10.6 | Page 30

Differentiate the following w.r.t. x : `cot^-1((a^2 - 6x^2)/(5ax))`

Exercise 1.2 | Q 10.7 | Page 30

Differentiate the following w.r.t. x : `tan^-1((a + btanx)/(b - atanx))`

Exercise 1.2 | Q 10.8 | Page 30

Differentiate the following w.r.t. x : `tan^-1((5 -x)/(6x^2 - 5x - 3))`

Exercise 1.2 | Q 10.9 | Page 30

Differentiate the following w.r.t. x : `cot^-1((4 - x - 2x^2)/(3x + 2))`

Exercise 1.3 [Pages 39 - 40]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Differentiation Exercise 1.3 [Pages 39 - 40]

Exercise 1.3 | Q 1.1 | Page 39

Differentiate the following w.r.t. x : `(x +  1)^2/((x + 2)^3(x + 3)^4`

Exercise 1.3 | Q 1.2 | Page 39

Differentiate the following w.r.t. x : `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`

Exercise 1.3 | Q 1.3 | Page 39

Differentiate the following w.r.t. x : `(x^2 + 3)^(3/2).sin^3 2x.2^(x^2)`

Exercise 1.3 | Q 1.4 | Page 39

Differentiate the following w.r.t. x : `((x^2 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x`

Exercise 1.3 | Q 1.5 | Page 39

Differentiate the following w.r.t. x : `(x^5.tan^3 4x)/(sin^2 3x)`

Exercise 1.3 | Q 1.6 | Page 39

Differentiate the following w.r.t. x : `x^(tan^(-1)x`

Exercise 1.3 | Q 1.7 | Page 39

Differentiate the following w.r.t. x : (sin x)x 

Exercise 1.3 | Q 1.8 | Page 39

Differentiate the following w.r.t. x : (sin xx)

Exercise 1.3 | Q 2.1 | Page 40

Differentiate the following w.r.t. x : xe + xx + ex + ee 

Exercise 1.3 | Q 2.2 | Page 40

Differentiate the following w.r.t. x : `x^(x^x) + e^(x^x)`

Exercise 1.3 | Q 2.3 | Page 40

Differentiate the following w.r.t. x : (logx)x – (cos x)cotx 

Exercise 1.3 | Q 2.4 | Page 40

Differentiate the following w.r.t. x : `x^(x^x) + (logx)^(sinx)`

Exercise 1.3 | Q 2.5 | Page 40

Differentiate the following w.r.t. x : etanx + (logx)tanx 

Exercise 1.3 | Q 2.6 | Page 40

Differentiate the following w.r.t. x : (sin x)tanx + (cos x)cotx 

Exercise 1.3 | Q 2.7 | Page 40

Differentiate the following w.r.t. x : `10^(x^(x)) + x^(x(10)) + x^(10x)`

Exercise 1.3 | Q 2.8 | Page 40

Differentiate the following w.r.t. x : `[(tanx)^(tanx)]^(tanx) "at"  x = pi/(4)`

Exercise 1.3 | Q 3.01 | Page 40

Find `"dy"/"dx"` if `sqrt(x) + sqrt(y) = sqrt(a)`

Exercise 1.3 | Q 3.02 | Page 40

Find `"dy"/"dx"` if `xsqrt(x) + ysqrt(y) = asqrt(a)`

Exercise 1.3 | Q 3.03 | Page 40

Find `"dy"/"dx"` if `x + sqrt(xy) + y ` = 1

Exercise 1.3 | Q 3.04 | Page 40

Find `"dy"/"dx"`If x3 + x2y + xy2 + y3 = 81

Exercise 1.3 | Q 3.05 | Page 40

Find `"dy"/"dx"` if `x^2y^2 - tan^-1(sqrt(x^2 + y^2)) = cot^-1(sqrt(x^2 + y^2))`

Exercise 1.3 | Q 3.06 | Page 40

Find `"dy"/"dx"` if xey + yex = 1

Exercise 1.3 | Q 3.07 | Page 40

Find `"dy"/"dx"` if ex+y = cos(x – y)

Exercise 1.3 | Q 3.08 | Page 40

Find `"dy"/"dx"` if cos (xy) = x + y

Exercise 1.3 | Q 3.09 | Page 40

Find `"dy"/"dx"` if `e^(e^(x - y)) = x/y`

Exercise 1.3 | Q 4.1 | Page 40

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : x7.y5 = (x + y)12 

Exercise 1.3 | Q 4.2 | Page 40

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : xpy4 = (x + y)p+4, p ∈ N

Exercise 1.3 | Q 4.3 | Page 40

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `sec((x^5 + y^5)/(x^5 - y^5))` = a2 

Exercise 1.3 | Q 4.4 | Page 40

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `tan^-1((3x^2 - 4y^2)/(3x^2 + 4y^2))` = a2 

Exercise 1.3 | Q 4.5 | Page 40

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `cos^-1((7x^4 + 5y^4)/(7x^4 - 5y^4)) = tan^-1a`

Exercise 1.3 | Q 4.6 | Page 40

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `log((x^20 - y^20)/(x^20 + y^20))` = 20

Exercise 1.3 | Q 4.7 | Page 40

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `e^((x^7 - y^7)/(x^7 + y^7)` = a

Exercise 1.3 | Q 4.8 | Page 40

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `sin((x^3 - y^3)/(x^3 + y^3))` = a3 

Exercise 1.3 | Q 5.01 | Page 40

If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.

Exercise 1.3 | Q 5.02 | Page 40

If `log_10((x^3 - y^3)/(x^3 + y^3)) = 2, "show that" "dy"/"dx" = -(99x^2)/(101y^2)`

Exercise 1.3 | Q 5.03 | Page 40

If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.

Exercise 1.3 | Q 5.04 | Page 40

If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.

Exercise 1.3 | Q 5.05 | Page 40

If `sin^-1((x^5 - y^5)/(x^5 + y^5)) = pi/(6), "show that" "dy"/"dx" = x^4/(3y^4)`

Exercise 1.3 | Q 5.06 | Page 40

If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.

Exercise 1.3 | Q 5.07 | Page 40

If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.

Exercise 1.3 | Q 5.08 | Page 40

If y = `sqrt(logx + log x + sqrt(log x + ... ∞)`, show that `"dy"/"dx" = (1)/(x(2y - 1)`.

Exercise 1.3 | Q 5.09 | Page 40

If y = `x^(x^(x^(.^(.^.∞))`, show that `"dy"/"dx" = y^2/(x(1 - logy).`.

Exercise 1.3 | Q 5.1 | Page 40

If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.

Exercise 1.4 [Pages 48 - 49]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Differentiation Exercise 1.4 [Pages 48 - 49]

Exercise 1.4 | Q 1.1 | Page 48

Find `"dy"/"dx"` if x = at2, y = 2at

Exercise 1.4 | Q 1.2 | Page 48

Find `"dy"/"dx"` if x = a cot θ, y = b cosec θ

Exercise 1.4 | Q 1.3 | Page 48

Find `"dy"/"dx"`, if : x = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`

Exercise 1.4 | Q 1.4 | Page 48

Find `"dy"/"dx"`, if : x = sinθ, y = tanθ

Exercise 1.4 | Q 1.5 | Page 48

Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)

Exercise 1.4 | Q 1.6 | Page 48

Find `"dy"/"dx"`, if : x = `(t + 1/t), y = a(t + 1/t)`, where a > 0, a ≠ 1, t ≠ 0.

Exercise 1.4 | Q 1.7 | Page 48

Find `"dy"/"dx"`, if : `x = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`

Exercise 1.4 | Q 1.8 | Page 48

Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.

Exercise 1.4 | Q 2.1 | Page 48

Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`

Exercise 1.4 | Q 2.2 | Page 48

Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`

Exercise 1.4 | Q 2.3 | Page 48

Find `"dy"/"dx"` if : x = t2 + t + 1, y = `sin((pit)/2) + cos((pit)/2) "at"  t = 1`

Exercise 1.4 | Q 2.4 | Page 48

Find `"dy"/"dx"` if : x = 2 cot t + cos 2t, y = 2 sin t – sin 2t at t = `pi/(4)`

Exercise 1.4 | Q 2.5 | Page 48

Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`

Exercise 1.4 | Q 3.1 | Page 48

If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "show that" "dy"/"dx" = y/x`.

Exercise 1.4 | Q 3.2 | Page 48

If x = esin3t, y = ecos3t, then show that `"dy"/"dx" = -(ylogx)/(xlogy)`.

Exercise 1.4 | Q 3.3 | Page 48

If x = `(t + 1)/(t - 1), y = (1 - t)/(t + 1), "then show that"  y^2 - "dy"/"dx"` = 0.

Exercise 1.4 | Q 3.4 | Page 48

If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^3`.

Exercise 1.4 | Q 3.5 | Page 48

If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.

Exercise 1.4 | Q 3.6 | Page 48

If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.

Exercise 1.4 | Q 3.7 | Page 48

If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + "dy"/dx"` = 0

Exercise 1.4 | Q 3.8 | Page 48

If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dy"/"dx" = -(b^2y)/(a^2x)`.

Exercise 1.4 | Q 4.1 | Page 49

DIfferentiate x sin x w.r.t. tan x.

Exercise 1.4 | Q 4.2 | Page 49

Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`

Exercise 1.4 | Q 4.3 | Page 49

Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.

Exercise 1.4 | Q 4.4 | Page 49

Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`

Exercise 1.4 | Q 4.5 | Page 49

Differentiate 3x w.r.t. logx3.

Exercise 1.4 | Q 4.6 | Page 49

DIfferentiate `tan^-1((cosx)/(1 + sinx)) w.r.t. sec^-1 x.`

Exercise 1.4 | Q 4.7 | Page 49

Differentiate xx w.r.t. xsix.

Exercise 1.4 | Q 4.8 | Page 49

Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t  tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.

Exercise 1.5 [Page 60]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Differentiation Exercise 1.5 [Page 60]

Exercise 1.5 | Q 1.1 | Page 60

Find the second order derivatives of the following : `2x^5 - 4x^3 - (2)/x^2 - 9`

Exercise 1.5 | Q 1.2 | Page 60

Find the second order derivatives of the following : e2x . tan x

Exercise 1.5 | Q 1.3 | Page 60

Find the second order derivatives of the following : e4x. cos 5x

Exercise 1.5 | Q 1.4 | Page 60

Find the second order derivatives of the following : x3.logx

Exercise 1.5 | Q 1.5 | Page 60

Find the second order derivatives of the following : log(logx)

Exercise 1.5 | Q 1.6 | Page 60

Find the second order derivatives of the following : xx 

Exercise 1.5 | Q 2.1 | Page 60

Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y =  a(1 – cos θ)

Exercise 1.5 | Q 2.2 | Page 60

Find `(d^2y)/(dx^2)` of the following : x = 2at2, y = 4at

Exercise 1.5 | Q 2.3 | Page 60

Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin3θ at θ = `pi/(2)`

Exercise 1.5 | Q 2.4 | Page 60

Find `(d^2y)/(dx^2)` of the following : x = a cos θ, y = b sin θ at θ = `pi/(4)`.

Exercise 1.5 | Q 3.01 | Page 60

If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.

Exercise 1.5 | Q 3.02 | Page 60

If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.

Exercise 1.5 | Q 3.03 | Page 60

If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.

Exercise 1.5 | Q 3.04 | Page 60

If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.

Exercise 1.5 | Q 3.05 | Page 60

If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.

Exercise 1.5 | Q 3.06 | Page 60

If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = m, "show"  (d^2y)/(dx^2)` = 0.

Exercise 1.5 | Q 3.07 | Page 60

If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 0.

Exercise 1.5 | Q 3.08 | Page 60

If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.

Exercise 1.5 | Q 3.09 | Page 60

If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.

Exercise 1.5 | Q 3.1 | Page 60

If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.

Exercise 1.5 | Q 3.11 | Page 60

If x2 + 6xy + y2 = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.

Exercise 1.5 | Q 3.12 | Page 60

If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^3 + y^2)/(y^3)`.

Exercise 1.5 | Q 4.01 | Page 60

Find the nth derivative of the following : (ax + b)m 

Exercise 1.5 | Q 4.02 | Page 60

Find the nth derivative of the following : `(1)/x`

Exercise 1.5 | Q 4.03 | Page 60

Find the nth derivative of the following : eax+b 

Exercise 1.5 | Q 4.04 | Page 60

Find the nth derivative of the following : apx+q 

Exercise 1.5 | Q 4.05 | Page 60

Find the nth derivative of the following : log (ax + b)

Exercise 1.5 | Q 4.06 | Page 60

Find the nth derivative of the following : cos x

Exercise 1.5 | Q 4.07 | Page 60

Find the nth derivative of the following : sin (ax + b)

Exercise 1.5 | Q 4.08 | Page 60

Find the nth derivative of the following : cos (3 – 2x)

Exercise 1.5 | Q 4.09 | Page 60

Find the nth derivative of the following : log (2x + 3)

Exercise 1.5 | Q 4.1 | Page 60

Find the nth derivative of the following : `(1)/(3x - 5)`

Exercise 1.5 | Q 4.11 | Page 60

Find the nth derivative of the following : y = eax . cos (bx + c)

Exercise 1.5 | Q 4.12 | Page 60

Find the nth derivative of the following : y = e8x . cos (6x + 7)

Miscellaneous Exercise 1 [Pages 61 - 63]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Differentiation Miscellaneous Exercise 1 [Pages 61 - 63]

Miscellaneous Exercise 1 | Q 1 | Page 61

Choose the correct option from the given alternatives : 

Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is 

  • `-(29)/(15)`

  • `(7)/(3)`

  • `(31)/(15)`

  • `(29)/(15)`

Miscellaneous Exercise 1 | Q 2 | Page 62

Choose the correct option from the given alternatives :

If y = sec (tan –1x), then `"dy"/"dx"` at x = 1, is equal to

  • `(1)/(2)`

  • 1

  • `(1)/sqrt(2)`

  • `sqrt(2)`

Miscellaneous Exercise 1 | Q 3 | Page 62

Choose the correct option from the given alternatives :

If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?

  • `(2.4^x.log4)/(1 + 4^(2x)` 

  • `(4^(x + 1).log2)/(1 + 4^(2x)`

  • `(4^(x + 1).log4)/(1 + 4^(4x)`

  • `(2^(2^((x + 1)).log2))/(1 + 2^(4x)`

Miscellaneous Exercise 1 | Q 4 | Page 62

Choose the correct option from the given alternatives :

If xy = yx, then `"dy"/"dx"` = ..........

  • `(x(xlogy - y))/(y(ylogx - x)`

  • `(y(xlogy - y))/(x(ylogx - x)`

  • `(y^2(1 - logx))/(x^2(1 - logy)`

  • `(y(1 - logx))/(x(1 - logy)`

Miscellaneous Exercise 1 | Q 5 | Page 62

Choose the correct option from the given alternatives :

If y = sin (2sin–1 x), then dx = ........

  • `(2 - 4x^2)/sqrt(1 - x^2)`

  • `(2 + 4x^2)/sqrt(1 - x^2)`

  • `(4x^2 - 1)/sqrt(1 - x^2)`

  • `(1 - 2x^2)/sqrt(1 - x^2)`

Miscellaneous Exercise 1 | Q 6 | Page 62

Choose the correct option from the given alternatives :

If y = `tan^-1(x/(1 + sqrt(1 - x^2))) + sin[2tan^-1(sqrt((1 - x)/(1 + x)))] "then" "dy"/"dx"` = ...........

  • `x/sqrt(1 - x^2)`

  • `(1 - 2x)/sqrt(1 - x^2)`

  • `(1 - 2x)/(2sqrt(1 - x^2)`

  • `(1 - 2x^2)/sqrt(1 - x^2)`

Miscellaneous Exercise 1 | Q 7 | Page 62

Choose the correct option from the given alternatives :

If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = ..........

  • 2

  • 0

  • –1

  • 1

Miscellaneous Exercise 1 | Q 8 | Page 62

Choose the correct option from the given alternatives :

If g is the inverse of function f and f'(x) = `(1)/(1 + x)`, then the value of g'(x) is equal to :

  • 1 + x7 

  • `(1)/(1 + [g(x)]^7`

  • 1 + [g(x)]7 

  • 7x6 

Miscellaneous Exercise 1 | Q 9 | Page 62

Choose the correct option from the given alternatives :

If `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........

  • `(1)/(1 + x)^2`

  • `-(1)/(1 + x)^2`

  • (1 + x)2 

  • `-x/(x + 1)`

Miscellaneous Exercise 1 | Q 10 | Page 62

Choose the correct option from the given alternatives :

If y `tan^-1(sqrt((a - x)/(a +  x)))`, where – a < x < a, then `"dy"/"dx"` = .........

  • `x/sqrt(a^2 - x^2)`

  • `a/sqrt(a^2 - x^2)`

  • `-(1)/(2sqrt(a^2 - x^2)`

  • `(1)/(2sqrt(a^2 - x^2)`

Miscellaneous Exercise 1 | Q 11 | Page 63

Choose the correct option from the given alternatives :

If x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........

  • `(8sqrt(2))/(api)`

  • `-(8sqrt(2))/(api)`

  • `(api)/(8sqrt(2))`

  • `(4sqrt(2))/(api)`

Miscellaneous Exercise 1 | Q 12 | Page 63

Choose the correct option from the given alternatives :

If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are

  • x2, – x, – y

  • x2, x, y

  • x2, x, – y

  • x2, –x, y

Miscellaneous Exercise 1 [Pages 63 - 64]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 1 Differentiation Miscellaneous Exercise 1 [Pages 63 - 64]

Miscellaneous Exercise 1 | Q 1 | Page 63

Solve the following : 

f(x) = –x, for – 2 ≤ x < 0
= 2x, for 0 ≤ x < 2
= `(18 - x)/(4)`, for 2 < x ≤ 7
g(x) = 6 – 3x, for 0 ≤ x < 2
= `(2x - 4)/(3)`, for 2 < x ≤ 7
Let u (x) = f[g(x)], v(x) = g[f(x)] and w(x) = g[g(x)]. Find each derivative at x = 1, if it exists i.e. find u'(1), v' (1) and w'(1). If it doesn't exist, then explain why?

Miscellaneous Exercise 1 | Q 2 | Page 63

Solve the following : 

The values of f(x), g(x), f'(x) and g'(x) are given in the following table :

x f(x) g(x) f'(x) fg'(x)
– 1 3 2 – 3 4
2 2 – 1 – 5 – 4

Match the following :

A Group – Function B Group – Derivative
(A)`"d"/"dx"[f(g(x))]"at" x = -1` 1.  – 16
(B)`"d"/"dx"[g(f(x) - 1)]"at" x = -1` 2.     20
(C)`"d"/"dx"[f(f(x) - 3)]"at" x = 2` 3.  – 20
(D)`"d"/"dx"[g(g(x))]"at"x = 2` 5.     12
Miscellaneous Exercise 1 | Q 3 | Page 63

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1: 

x f(x) g(x) f')x) g'(x)
0 1   5 `(1)/(3)`
1 3 – 4 `-(1)/(3)` `-(8)/(3)`

(i) The derivative of f[g(x)] w.r.t. x at x = 0 is ......
(ii) The derivative of g[f(x)] w.r.t. x at x = 0 is ......
(iii) The value of `["d"/"dx"[x^(10) + f(x)]^(-2)]_(x = 1_` is ........
(iv) The derivative of f[(x + g(x))] w.r.t. x at x = 0 is ...

Miscellaneous Exercise 1 | Q 4.1 | Page 64

Differentiate the following w.r.t. x : `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`

Miscellaneous Exercise 1 | Q 4.2 | Page 64

Differentiate the following w.r.t. x : `sin^2[cot^-1(sqrt((1 + x)/(1 - x)))]`

Miscellaneous Exercise 1 | Q 4.3 | Page 64

Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`

Miscellaneous Exercise 1 | Q 4.4 | Page 64

Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`

Miscellaneous Exercise 1 | Q 4.5 | Page 64

Differentiate the following w.r.t. x : `tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`

Miscellaneous Exercise 1 | Q 4.6 | Page 64

Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`

Miscellaneous Exercise 1 | Q 5.1 | Page 64

If `sqrt(y + x) + sqrt(y - x)` = c, show that `"dy"/"dx" = y/x - sqrt(y^2/x^2 - 1)`.

Miscellaneous Exercise 1 | Q 5.2 | Page 64

If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.

Miscellaneous Exercise 1 | Q 5.3 | Page 64

If x sin (a + y) + sin a . cos (a + y) = 0, then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.

Miscellaneous Exercise 1 | Q 5.4 | Page 64

If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.

Miscellaneous Exercise 1 | Q 5.5 | Page 64

If `x = e^(x/y)`, then show that `"dy"/"dx" = (x - y)/(xlogx)`

Miscellaneous Exercise 1 | Q 5.6 | Page 64

If y = f(x) is a differentiable function of x, then show that `(d^2x)/(dy^2) = -(dy/dx)^-3.("d^2y)/(dx^2)`.

Miscellaneous Exercise 1 | Q 6.1 | Page 64

DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. tan^-1(sqrt((2xsqrt(1 + x^2))/(1 - 2x^2)))`.

Miscellaneous Exercise 1 | Q 6.2 | Page 64

Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).

Miscellaneous Exercise 1 | Q 6.3 | Page 64

DIfferentiate `tan^-1((sqrt(1 + x^2) - 1)/x) w.r.t. cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2)))).`

Miscellaneous Exercise 1 | Q 7.1 | Page 64

If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`

Miscellaneous Exercise 1 | Q 7.2 | Page 64

If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.

Miscellaneous Exercise 1 | Q 7.3 | Page 64

If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.

Miscellaneous Exercise 1 | Q 7.4 | Page 64

If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.

Miscellaneous Exercise 1 | Q 7.5 | Page 64

If y = Aemx + Benx, show that y2 – (m + n)y1 + mny = 0.

Chapter 1: Differentiation

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Miscellaneous Exercise 1
Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board - Shaalaa.com

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 1 - Differentiation

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 1 (Differentiation) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 1 Differentiation are Differentiation, Derivatives of Composite Functions - Chain Rule, Geometrical Meaning of Derivative, Derivatives of Inverse Functions, Logarithmic Differentiation, Derivatives of Implicit Functions, Derivatives of Parametric Functions, Higher Order Derivatives.

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