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

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

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

Question 1.
Find the derivative of x3 from the first principle. [Mar. ’15 (TS), ’98]
Solution:
Let f(x) = x3 then f(x + h) = (x + h)3
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q1

Question 2.
Find the derivative of \(\sqrt{x+1}\) from the first principle. [Mar. ’12, ’05]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q2
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q2.1

TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type

Question 3.
Find the derivative of sin 2x from the first principle. [B.P. May ’15 (TS), ’10, ’91; Mar. ’02; Mar. ’18 (AP)]
Solution:
Let f(x) = sin 2x
f(x + h) = sin 2(x + h) = sin 2x + 2h
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q3
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q3.1

Question 4.
Find the derivative of cos ax from the first principle. [May ’14; Mar. ’13 (old), ’13, ’11, ’09]
Solution:
Let f(x) = cos ax
f(x + h) = cos a(x + h) = cos(ax + ah)
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q4

Question 5.
Find the derivative of tan 2x from the first principle. [Mar. ’14, ’13 (old). ’05; May ’13. ’11; May ’15 (AP)]
Solution:
Let f(x) = tan 2x
f(x + h) = tan 2(x + h) = tan (2x + 2h)
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q5
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q5.1

Question 6.
Find the derivative of sec 3x from the first principle. [Mar. ’16 (AP), ’12, ’08]
Solution:
Let f(x) = sec 3x
f(x + h) = sec 3(x + h) = sec (3x + 3h)
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q6
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q6.1

Question 7.
Find the derivative of x sin x from the first principle. [Mar. ’18, ’15 (AP), ’10; May ’09]
Solution:
Let f(x) = x sin x
f(x + h) = (x + h) sin (x + h)
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q7
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q7.1

TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type

Question 8.
Find the derivative of cos2x from the first principle. [Mar. ’19 (TS); May ’08, ’04]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q8

Question 9.
Find the derivative of log x from the first principle. [Mar. ’03]
Solution:
Given, f(x) = log x
Now, f(x + h) = log (x + h)
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q9
∴ f is differentiable at x and f'(x) = \(\frac{1}{x}\)

Question 10.
Prove that \(\frac{d}{d x} \mathbf{u v}=\mathbf{u} \frac{d v}{d x}+v \frac{d u}{d x}\). [May ’97]
(Or)
If f, g are two differentiable functions at x then fg is differentiable at x. then show that (fg)’ (x) = f(x) g'(x) + g(x) f'(x).
Solution:
Since f and g are two differentiable functions at x, f'(x) and g'(x) exist.
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q10
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q10.1
= f(x + 0) . g'(x) + g(x) . f'(x)
= f(x) . g'(x) + g(x) . f'(x)
∴ fg is differentiable at x and (fg)’ (x) = f(x) g'(x) + g(x) f'(x).

Question 11.
Prove that \(\frac{d}{d x}\left(\frac{u}{v}\right)=\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^2}\). [May ’04, ’98]
(Or)
If f, g are two differentiable functions at x and g(x) ≠ 0 then \(\frac{f}{g}\) is differentiable at x, then show that \(\left(\frac{f}{g}\right)^{\prime}(x)=\frac{g(x) f^{\prime}(x)-f(x) g^{\prime}(x)}{[g(x)]^2}\)
Solution:
Since f, g are differentiable at x and f'(x), g'(x) exists.
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q11
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q11.1

Question 12.
Find the derivative of \(\cos ^{-1}\left(\frac{b+a \cos x}{a+b \cos x}\right)\), (a > 0, b > 0). [May ’09]
Solution:
Let y = \(\cos ^{-1}\left(\frac{b+a \cos x}{a+b \cos x}\right)\)
Differentiating on both sides with respect to ‘r’ to ‘x’
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q12
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q12.1

Question 13.
If xy = ex-y, then show that \(\frac{d y}{d x}=\frac{\log x}{(1+\log x)^2}\). [Mar. ’08, ’07, ’96, ’88; May ’00, ’95]
Solution:
Given, xy = ex-y
Taking logarithms on both sides,
log(xy) = log(ex-y)
y log x = (x – y) log e
y log x = x – y
y + y log x = x
y(1 + log x) = x
y = \(\frac{x}{1+\log x}\)
Differentiating on both sides with respect to ‘x’
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q13

TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type

Question 14.
If sin y = x sin (a + y), then show that \(\frac{d y}{d x}=\frac{\sin ^2(a+y)}{\sin a}\). [Mar. ’95, May ’87]
Solution:
Given, sin y = x sin (a + y)
x = \(\frac{\sin y}{\sin (a+y)}\)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q14
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q14.1

Question 15.
Differentiate \(\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\) with respect to \(\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)\). [Mar. ’13 (Old); May ’04, ’95]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q15
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q15.1

Question 16.
Find the derivative of tan-1(sec x + tan x). [May ’97]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q16
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q16.1

Question 17.
If x = a (cos t + t sin t), y = a (sin t – t cos t) then find \(\frac{d \mathbf{y}}{d \mathbf{x}}\). [Mar. ’16 (TS), May ’08, ’00, ’93]
Solution:
Given that x = a(cos t + t sin t)
Differentiating on both sides with respect to ‘t’.
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q17

Question 18.
If x = a(t – sin t), y = a(1 + cos t) then find \(\frac{d^2 \mathbf{y}}{\mathbf{d x}^2}\). [May ’02]
Solution:
Given, x = a(t – sin t)
Differentiating on both sides with respect to ‘x’.
\(\frac{\mathrm{dx}}{\mathrm{dt}}\) = a(1 – cos t)
y = a(1 + cos t)
Differentiating on both sides with respect to ‘x’.
\(\frac{\mathrm{dy}}{\mathrm{dt}}\) = a(0 – sin t) = -a sin t
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q18

Question 19.
Find the second order derivative of \(\tan ^{-1}\left(\frac{1+x}{1-x}\right)\). [May ’12]
Solution:
Given, f(x) = \(\tan ^{-1}\left(\frac{1+x}{1-x}\right)\)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q19
Again differentiating on both sides with respect to ‘x’.
f”(x) = \(-\frac{1}{\left(1+x^2\right)^2}(0+2 x)=\frac{-2 x}{\left(1+x^2\right)^2}\)

TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type

Question 20.
If y = aenx + be-nx, then prove that y” = n2y. [Mar. ’15 (AP); May ’14]
Solution:
Given y = aenx + be-nx ………(1)
Differentiating on both sides with respect to ‘x’.
y’ = aenx (n) + be-nx (-n)
y’ = n aenx – n be-nx
again differentiating on both sides with respect to ‘x’
y” = na . enx (n) – n be-nx (-n)
= n2 aexnx + n2be-nx
= n2 (aenx + be-nx)
= n2y (∵ from 1)
∴ y” = n2y

Question 21.
If y = axn+1 + bx-n then prove that x2y11 = n(n + 1)y. [May ’10; Mar. ’06]
Solution:
Given, y = axn+1 + bx-n …….(1)
Differentiating on both sides with respect to ‘x’.
y1 = a . (n + 1) xn+1-1 + b(-n) x-n-1
= a(n + 1)xn – bn . x-n-1
Again differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q21

Question 22.
If y = a cos x + (b + 2x) sin x, then prove that y11 + y = 4 cos x. [May ’07]
Solution:
Given, y = a cos x + (b + 2x) sin x ………(1)
Differentiating on both sides with respect to ‘x’.
y1 = a(-sin x) + (b + 2x) cos x + sin x (0 + 2 . 1)
y1 = -a sin x + (b + 2x) cos x + 2 sin x
Again differentiating of both sides with respect to ‘x’.
y11 = -a cos x + (b + 2x) (-sin x) + cos x (0 + 2 . 1) + 2 cos x
= -a cos x – (b + 2x) sin x + 2 cos x + 2 cos x
= -a cos x – (b + 2x) sin x + 4 cos x
= -[a cos x + (b + 2x) sin x] + 4 cos x
y11 = -y + 4 cos x [∵ from (1)]
y11 + y = 4 cos x

Question 23.
If ax2 + 2hxy + by2 = 1 then prove that \(\frac{d^2 y}{d x^2}=\frac{h^2-a b}{(h x+b y)^3}\). [Mar. ’08; May ’97]
Solution:
Given, ax2 + 2hxy + by2 = 1 ……..(1)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q23
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q23.1

TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type

Question 24.
Find the derivative of cot x from the first principle. [Mar. ’19, ’17 (AP)]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Short Answer Type Q24

Leave a Comment