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## Chapter 1: Differentiation

### 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]

**Differentiate the following w.r.t.x: **

(x^{3} – 2x – 1)^{5}

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

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

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

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

**Differentiate the following w.r.t.x:**

`(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`

Differentiate the following w.r.t.x: cos(x^{2} + a^{2})

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

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

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

Differentiate the following w.r.t.x: cot^{3}[log(x^{3})]

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

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

Differentiate the following w.r.t.x: log[cos(x^{3} – 5)]

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

Differentiate the following w.r.t.x: cos^{2}[log(x^{2} + 7)]

**Differentiate the following w.r.t.x:**

tan[cos(sinx)]

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

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

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

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

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

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

Differentiate the following w.r.t.x: sin^{2}x^{2} – cos^{2}x^{2}

**Differentiate the following w.r.t.x:**

(x^{2 }+ 4x + 1)^{3} + (x^{3}− 5x − 2)^{4}

Differentiate the following w.r.t.x: (1 + 4x)^{5} (3 + x −x2)^{8 }

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

**Differentiate the following w.r.t.x: **

`(x^3 - 5)^5/(x^3 + 3)^3`

Differentiate the following w.r.t.x: (1 + sin^{2} x)^{2} (1 + cos^{2} x)^{3}

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

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

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

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

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

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

Differentiate the following w.r.t.x: log[tan^{3}x.sin^{4}x.(x^{2} + 7)^{7}]

**Differentiate the following w.r.t.x: **

`log(sqrt((1 - cos3x)/(1 + cos3x)))`

**Differentiate the following w.r.t.x:**

`log(sqrt((1 + cos((5x)/2))/(1 - cos((5x)/2))))`

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

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

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

Differentiate the following w.r.t.x:

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

**Differentiate the following w.r.t.x:**

y = (25)^{log5(secx)} − (16)^{log4(tanx)}

**Differentiate the following w.r.t.x:**

`(x^2 + 2)^4/(sqrt(x^2 + 5)`

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).

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).

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).

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).

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

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

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.

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

"Let f(x) = x^{2 }+ 5 and g (x) = e^{x} + 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}`

### 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]

**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)`

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)`

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)`

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)

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

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f^{–1}(y) in the following: y = e^{x} – 3

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f^{–1}(y) in the following: y = e^{2x-3}

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)`

Find the derivative of the inverse function of the following : y = x^{2}·e^{x}

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

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

Find the derivative of the inverse function of the following : y = x^{2} + log x

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

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = x^{5} + 2x^{3} + 3x, at x = 1

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

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

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) + x^{2}

If f(x) = x^{3} + x – 2, find (f^{–1})'(0).

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

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

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

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

Differentiate the following w.r.t. x : cot^{–1}(x^{3})

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

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

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

Differentiate the following w.r.t. x : cos^{–1}(1 –x^{2})

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

Differentiate the following w.r.t. x : cos^{3}[cos^{–1}(x^{3})]

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

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

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

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

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

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

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

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

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

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

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

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

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)))]`

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

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

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

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

Differentiate the following w.r.t. x :

`cos^-1[(3cos(e^x) + 2sin(e^x))/sqrt(13)]`

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

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

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

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

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

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

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

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

Differentiate the following w.r.t. x :

`sin^-1(4^(x + 1/2)/(1 + 2^(4x)))`

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

Differentiate the following w.r.t. x :

`sin^(−1) ((1 − x^3)/(1 + x^3))`

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

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

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

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

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

Differentiate the following w.r.t. x :

`tan^(−1)[(2^(x + 2))/(1 − 3(4^x))]`

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

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

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

Differentiate the following w.r.t. x :

`tan^-1((5 -x)/(6x^2 - 5x - 3))`

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

### 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]

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

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

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

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

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

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

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

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

Differentiate the following w.r.t. x: x^{e} + x^{x} + e^{x }+ e^{e}

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

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

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

Differentiate the following w.r.t. x : e^{tanx} + (logx)^{tanx }

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

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

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

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

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

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

Find `"dy"/"dx"`If x^{3} + x^{2}y + xy^{2} + y^{3} = 81

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

Find `"dy"/"dx"` if xe^{y} + ye^{x} = 1

Find `"dy"/"dx"` if e^{x+y} = cos(x – y)

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

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

Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : x^{7}.y^{5} = (x + y)^{12}

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

Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `sec((x^5 + y^5)/(x^5 - y^5))` = a^{2}

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))` = a^{2}

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`

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

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

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

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

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

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

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

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

If x^{y} = e^{x–y}, then show that `"dy"/"dx" = logx/(1 + logx)^2`.

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

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

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

If e^{y} = y^{x}, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.

### 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]

Find `"dy"/"dx"` if x = at^{2}, y = 2at

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

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

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

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

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

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

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

Find `"dy"/"dx"` if : x = cosec^{2}θ, y = cot^{3}θ at θ= `pi/(6)`

Find `"dy"/"dx"` if : x = a cos^{3}θ, y = a sin^{3}θ at θ = `pi/(3)`

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

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

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

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

If x = e^{sin3t}, y = e^{cos3t}, then show that `"dy"/"dx" = -(ylogx)/(xlogy)`.

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

If x = a cos^{3}t, y = a sin^{3}t, show that `"dy"/"dx" = -(y/x)^3`.

If x = 2cos^{4}(t + 3), y = 3sin^{4(}t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.

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

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

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

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

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

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

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

Differentiate 3^{x }w.r.t. log_{x}3.

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

Differentiate x^{x} w.r.t. x^{six}.

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

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

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

Find the second order derivatives of the following : e^{2x} . tan x

Find the second order derivatives of the following : e^{4x}. cos 5x

Find the second order derivatives of the following : x^{3}.logx

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

Find the second order derivatives of the following : x^{x}

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

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

Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin^{3}θ at θ = `pi/(2)`

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

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

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

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

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

If y = e^{ax}.sin(bx), show that y_{2} – 2ay_{1} + (a^{2} + b^{2})y = 0.

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

If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x^{2} – 1)y_{2} + 4xy_{1} – y = 0.

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

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

If y = log (log 2x), show that xy_{2} + y_{1} (1 + xy_{1}) = 0.

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

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)`.

Find the n^{th} derivative of the following : (ax + b)^{m}

Find the n^{th} derivative of the following : `(1)/x`

Find the n^{th} derivative of the following : e^{ax+b}

Find the nth derivative of the following : a^{px+q}

Find the n^{th} derivative of the following : log (ax + b)

Find the n^{th} derivative of the following : cos x

Find the n^{th} derivative of the following : sin (ax + b)

Find the n^{th} derivative of the following : cos (3 – 2x)

Find the n^{th} derivative of the following : log (2x + 3)

Find the n^{th} derivative of the following : `(1)/(3x - 5)`

Find the n^{th} derivative of the following : y = e^{ax} . cos (bx + c)

Find the n^{th} derivative of the following : y = e^{8x} . cos (6x + 7)

### 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]

**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)`

**Choose the correct option from the given alternatives :**

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

`(1)/(2)`

1

`(1)/sqrt(2)`

`sqrt(2)`

**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)`

**Choose the correct option from the given alternatives :**

If x^{y} = y^{x}, 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)`

**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)`

**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)`

**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

**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 + x

^{7}`(1)/(1 + [g(x)]^7`

1 + [g(x)]

^{7}7x

^{6}

**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)`

**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)`

**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)`

**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

x

^{2}, – x, – yx

^{2}, x, yx

^{2}, x, – yx

^{2}, –x, y

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

**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?

**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 |

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 ...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

If y^{2} = a^{2}cos^{2}x + b^{2}sin^{2}x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`

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

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

If y = A cos (log x) + B sin (log x), show that x^{2}y^{2} + xy_{1} + y = 0.

If y = Ae^{mx} + Be^{nx}, show that y_{2} – (m + n)y_{1} + mny = 0.

## Chapter 1: Differentiation

## 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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