Students must practice these Maths 1B Important Questions TS Inter 1st Year Maths 1B Differentiation Important Questions Very Short Answer Type to help strengthen their preparations for exams.

## TS Inter 1st Year Maths 1B Differentiation Important Questions Very Short Answer Type

Question 1.

Find the derivative of f(x) = e^{x} (x^{2} + 1). [May ’02]

Solution:

Given f(x) = e^{x} (x^{2} + 1)

Let y = e^{x} (x^{2} + 1)

Differentiating with respect to x on both sides

Question 2.

If f(x) = x^{2} . 2^{x} . log x (x > 0), find f'(x). [May ’10]

Solution:

Given f(x) = x^{2} . 2^{x} . log x

Let y = x^{2} . 2^{x} . log x

Differentiating with respect to x on both sides

Question 3.

If f(x) = \(7^{x^3+3 x}\) (x > 0), then find f'(x). [Mar. ’17 (TS); May ’05]

Solution:

Question 4.

If y = e^{2x} log(3x + 4) then find \(\frac{d y}{d x}\). [May ’13; Mar. ’13 (Old)]

Solution:

Given, f(x) = e^{2x} log(3x + 4)

Let y = e^{2x} log(3x + 4)

Differentiating on both sides with respect to x

\(\frac{d y}{d x}=\frac{d}{d x}\left[e^{2 x} \log (3 x+4)\right]\)

Question 5.

If y = \(\frac{\mathbf{a x}+\mathbf{b}}{\mathbf{c x}+\mathbf{d}}\) then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\).

Solution:

Given, f(x) = \(\frac{\mathbf{a x}+\mathbf{b}}{\mathbf{c x}+\mathbf{d}}\)

Let y = \(\frac{\mathbf{a x}+\mathbf{b}}{\mathbf{c x}+\mathbf{d}}\)

Differentiating on both sides with respect to x

Question 6.

If f(x) = \(\mathbf{a}^{\mathbf{x}} \cdot \mathrm{e}^{\mathbf{x}^2}\), then find f'(x). [May ’08; B.P.]

Solution:

Given, f(x) = \(\mathbf{a}^{\mathbf{x}} \cdot \mathrm{e}^{\mathbf{x}^2}\)

Let y = \(\mathbf{a}^{\mathbf{x}} \cdot \mathrm{e}^{\mathbf{x}^2}\)

Differentiating on both sides with respect to x

Question 7.

If f(x) = log(sec x + tan x), find f'(x). [Mar. ’14; May ’11]

Solution:

Given, f(x) = log(sec x + tan x)

Differentiating on both sides with respect to x

Question 8.

If f(x) = 1 + x + x^{2} + ……….. + x^{100}, then find f'(1). [Mar. ’19 (TS); May ’14]

Solution:

Given f(x) = 1 + x + x^{2} + ……… + x^{100}

Differentiating on both sides with respect to ‘x’.

Question 9.

If y = \(\sin ^{-1} \sqrt{x}\), find \(\frac{d \mathbf{y}}{d x}\). [Mar. ’13]

Solution:

Given, y = \(\sin ^{-1} \sqrt{x}\)

Differentiating on both sides with respect to x

Question 10.

If y = sec(√tan x), find \(\frac{d \mathbf{y}}{d x}\). [May ’07]

Solution:

Given, y = sec(√tan x)

Differentiating on both sides with respect to x

Question 11.

If y = log(cosh 2x), find \(\frac{d \mathbf{y}}{d x}\). [Mar. ’12]

Solution:

Given y = log(cosh 2x)

Differentiating on both sides with respect to x

Question 12.

If y = log(sin(log x)), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).

Solution:

Given, y = log(sin(log x))

Differentiating on both sides with respect to ‘x’.

Question 13.

If y = \(\left(\cot ^{-1} x^3\right)^2\), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\). [May ’13, ’09; Mar. ’18 (TS)]

Solution:

Given y = \(\left(\cot ^{-1} x^3\right)^2\)

Differentiating on both sides with respect to ‘x’.

Question 14.

Find the derivative of log(tan 5x). [Mar. ’08]

Solution:

Given, y = log(tan 5x)

Differentiating on both sides with respect to ‘x’.

Question 15.

Find the derivative of \(\sinh ^{-1}\left(\frac{3 x}{4}\right)\). [May ’13 (Old)]

Solution:

Let y = \(\sinh ^{-1}\left(\frac{3 x}{4}\right)\)

Differentiating on both sides with respect to ‘x’.

Question 16.

Find the derivative of \(\log \left(\frac{x^2+x+2}{x^2-x+2}\right)\). [May ’06]

Solution:

Let y = \(\log \left(\frac{x^2+x+2}{x^2-x+2}\right)\)

Differentiating on both sides with respect to ‘x’.

\(\frac{d y}{d x}=\frac{d}{d x} \log \left(\frac{x^2+x+2}{x^2-x+2}\right)\)

Question 17.

Find the derivative of \(\log \left[\sin ^{-1}\left(e^x\right)\right]\). [Mar. ’10]

Solution:

Let y = \(\log \left[\sin ^{-1}\left(e^x\right)\right]\)

Differentiating on both sides with respect to ‘x’.

Question 18.

Find the derivation of x = tan(e^{-y}) with respect to x. [Mar. ’17 (TS), ’05; May ’03]

Solution:

Given, x = tan(e^{-y})

⇒ tan^{-1}x = e^{-y}

Differentiating on both sides with respect to ‘x’.

Question 19.

Find the derivative of cos[log(cot x)]. [Mar. ’13 (old)]

Solution:

Let y = cos[log(cot x)]

Differentiating on both sides with respect to ‘x’.

Question 20.

If y = \(\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\), then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [Mar. ’15 (AP), ’04; May ’98, ’92]

Solution:

Question 21.

If y = x^{x} (x > 0), find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [Mar. ’11; May ’97, ’96]

Solution:

Given, y = x^{x}

Taking logarithms on both sides,

log y = log x^{x}

log y = x log x

Differentiating on both sides with respect to ‘x’.

Question 22.

If x = a cos^{3}t, y = a sin^{3}t, find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [Mar. ’16 (AP), ’12, ’07, ’02; May ’12, ’11]

Solution:

Question 23.

If x^{3} + y^{3} – 3axy = 0, find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [Mar. ’00]

Solution:

Given, x^{3} + y^{3} – 3axy = 0

Differentiating on both sides with respect to ‘x’.

Question 24.

Find the derivative of sin^{-1}(3x – 4x^{3}) with respect to ‘x’. [Mar. ’16 (TS), May ’11, ’97]

Solution:

Let y = sin^{-1}(3x – 4x^{3})

Put x = sin θ

⇒ θ = sin^{-1}x

Now, y = sin^{-1}(3 sin θ – 4 sin^{3}θ)

= sin^{-1}(sin 3θ)

= 3θ

y = 3 sin^{-1}x

Differentiating on both sides with respect to ‘x’.

\(\frac{\mathrm{dy}}{\mathrm{dx}}=3 \frac{\mathrm{d}}{\mathrm{dx}} \sin ^1 \mathrm{x}\)

= \(3 \frac{1}{\sqrt{1-x^2}}\)

= \(\frac{3}{\sqrt{1-x^2}}\)

Question 25.

Find the derivative of \(\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)\). [May ’15 (TS); Mar. ’15 (TS), ’12, ’98]

Solution:

Question 26.

Find the derivative of \(\tan ^{-1} \sqrt{\frac{1-\cos x}{1+\cos x}}\). [May ’13 (old); May ’02]

Solution:

Question 27.

Find the derivative of \(\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)\). [Mar. ’17 (AP), ’13]

Solution:

Let y = \(\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)\)

Question 28.

If x = 3 cos t – 2 cos^{3}t, y = 3 sin t – 2 sin^{3}t, then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\).

Solution:

Question 29.

Find the derivative of y = x^{y}. [Mar. ’04, ’00, ’99]

Solution:

Given, y = x^{y}

Taking logarithms on both sides, we get

log y = log x^{y}

⇒ log y = y log x

Differentiating on both sides with respect to ‘x’.

\(\frac{d}{d x}(\log y)=\frac{d}{d x}(y \log x)\)

Question 30.

Find the derivative of ex with respect to √x. [Mar. ’03]

Solution:

Given, f(x) = e^{x}, g(x) = √x

Let u = e^{x}

Differentiating on both sides with respect to ‘x’.

\(\frac{d u}{d x}=\frac{d}{d x} e^x=e^x\)

Let v = √x

Differentiating on both sides with respect to ‘x’.

Question 31.

If y = \(\frac{2 x+3}{4 x+5}\), then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [May ’15 (AP)]

Solution:

Question 32.

Find the derivative of y = \(\sqrt{2 x-3}+\sqrt{7-3 x}\). [Mar. ’15 (TS)]

Solution:

Given y = \(\sqrt{2 x-3}+\sqrt{7-3 x}\)

Question 33.

Find the derivative of 5 sin x + e^{x} log x. [Mar. ’17 (AP)]

Solution:

Let y = 5 sin x + e^{x} log x

Differentiating on both sides with respect to x