Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 1 Differentiation Ex 1.5 Questions and Answers.

## Maharashtra State Board 12th Maths Solutions Chapter 1 Differentiation Ex 1.5

Question 1.

Find the second order derivatives of the following:

(i) 2x^{5} – 4x^{3} – \(\frac{2}{x^{2}}\) – 9

Solution:

Let y = 2x^{5} – 4x^{3} – \(\frac{2}{x^{2}}\) – 9

(ii) e^{2x} . tan x

Solution:

Let y = e^{2x} . tan x

(iii) e^{4x} . cos 5x

Solution:

Let y = e^{4x} . cos 5x

(iv) x^{3} . log x

Solution:

Let y = x^{3} . log x

(v) log(log x)

Solution:

Let y = log(log x)

(vi) x^{x}

Solution:

y = x^{x}

log y = log x^{x} = x log x

Differentiating both sides w.r.t. x, we get

Question 2.

Find \(\frac{d^{2} y}{d x^{2}}\) of the following:

(i) x = a(θ – sin θ), y = a (1 – cos θ)

Solution:

x = a(θ – sin θ), y = a (1 – cos θ)

Differentiating x and y w.r.t. θ, we get

\(\frac{d x}{d \theta}=a \frac{d}{d \theta}(\theta-\sin \theta)\) = a(1 – cos θ) …….(1)

(ii) x = 2at^{2}, y = 4at

Solution:

x = 2at^{2}, y = 4at

Differentiating x and y w.r.t. t, we get

(iii) x = sin θ, y = sin^{3}θ at θ = \(\frac{\pi}{2}\)

Solution:

x = sin θ, y = sin^{3}θ

Differentiating x and y w.r.t. θ, we get,

(iv) x = a cos θ, y = b sin θ at θ = \(\frac{\pi}{4}\)

Solution:

x = a cos θ, y = b sin θ

Differentiating x and y w.r.t. θ, we get

Question 3.

(i) If x = at^{2} and y = 2at, then show that \(x y \frac{d^{2} y}{d x^{2}}+a=0\)

Solution:

x = at^{2}, y = 2at ………(1)

Differentiating x and y w.r.t. t, we get

(ii) If y = \(e^{m \tan ^{-1} x}\), show that \(\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+(2 x-m) \frac{d y}{d x}=0\)

Solution:

y = \(e^{m \tan ^{-1} x}\) ……..(1)

(iii) If x = cos t, y = e^{mt}, show that \(\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-m^{2} y=0\)

Solution:

x = cos t, y = e^{mt}

(iv) If y = x + tan x, show that \(\cos ^{2} x \cdot \frac{d^{2} y}{d x^{2}}-2 y+2 x=0\)

Solution:

y = x + tan x

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

Solution:

y = e^{ax} . sin (bx) ………(1)

(vi) If \(\sec ^{-1}\left(\frac{7 x^{3}-5 y^{3}}{7 x^{3}+5 y^{3}}\right)=m\), show that \(\frac{d^{2} y}{d x^{2}}=0\)

Solution:

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

Solution:

2y = \(\sqrt{x+1}+\sqrt{x-1}\) …… (1)

Differentiating both sides w.r.t. x, we get

(viii) If y = \(\log \left(x+\sqrt{x^{2}+a^{2}}\right)^{m}\), show that \(\left(x^{2}+a^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=0\)

Solution:

y = \(\log \left(x+\sqrt{x^{2}+a^{2}}\right)^{m}\) = \(m \log \left(x+\sqrt{x^{2}+a^{2}}\right)\)

(ix) If y = sin(m cos^{-1}x), then show that \(\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+m^{2} y=0\)

Solution:

y = sin(m cos^{-1}x)

sin^{-1}y = m cos^{-1}x

Differentiating both sides w.r.t. x, we get

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

Solution:

y = log(log 2x)

(xi) If x^{2} + 6xy + y^{2} = 10, show that \(\frac{d^{2} y}{d x^{2}}=\frac{80}{(3 x+y)^{3}}\)

Solution:

x^{2} + 6xy + y^{2} = 10 …… (1)

Differentiating both sides w.r.t. x, we get

(xii) If x = a sin t – b cos t, y = a cos t + b sin t, Show that \(\frac{d^{2} y}{d x^{2}}=-\frac{x^{2}+y^{2}}{y^{3}}\)

Solution:

x = a sin t – b cos t, y = a cos t + b sin t

Differentiating x and y w.r.t. t, we get

Question 4.

Find the nth derivative of the following:

(i) (ax + b)^{m}

Solution:

Let y = (ax + b)^{m}

(ii) \(\frac{1}{x}\)

Solution:

Let y = \(\frac{1}{x}\)

(iii) e^{ax+b}

Solution:

Let y = e^{ax+b}

(iv) a^{px+q}

Solution:

Let y = a^{px+q}

(v) log(ax + b)

Solution:

Let y = log(ax + b)

Then \(\frac{d y}{d x}=\frac{d}{d x}[\log (a x+b)]\)

(vi) cos x

Solution:

Let y = cos x

(vii) sin(ax + b)

Solution:

Let y = sin(ax + b)

(viii) cos(3 – 2x)

Solution:

(ix) log(2x + 3)

Solution:

(x) \(\frac{1}{3 x-5}\)

Solution:

Let y = \(\frac{1}{3 x-5}\)

(xi) y = e^{ax} . cos (bx + c)

Solution:

y = e^{ax} . cos (bx + c)

(xii) y = e^{8x} . cos (6x + 7)

Solution: