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

## 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

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

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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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