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

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

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

Question 1.
If y = \(\tan ^{-1}\left[\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right]\), then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [Mar. ’18 (TS); Mar. ’16 (AP), ’12, ’10, ’09, ’04; May ’15 (AP & TS), ’12, ’97]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q1
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q1.1
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q1.2

Question 2.
If y = xtan x + (sin x)cos x, then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [Mar. ’14, ’13 (Old), ’11, ’08, ’07; May ’13, ’06; Mar. ’18 (AP)]
Solution:
Given y = xtan x + (sin x)cos x
Differentiating on both sides with respect to ‘x’.
Let u = xtan x
v = sin xcos x
y = u + v
\(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{u}+\mathrm{v})\)
\(\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}\) ……(1)
Now, u = xtan x
Taking logarithms on both sides,
log u = log xtan x
log u = tan x log x
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q2
Now, v = (sin x)cos x
Taking logarithms on both sides,
log v = log (sin x)cos x
log v = cos x log(sin x)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q2.1

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

Question 3.
If x = \(\frac{3 a t}{1+t^3}\), y = \(\frac{3 \mathrm{at}^2}{1+\mathrm{t}^3}\), then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [B.P.]
Solution:
Given that x = \(\frac{3 a t}{1+t^3}\)
Differentiating on both sides with respect to ‘t’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q3
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q3.1

Question 4.
If \(\sqrt{1-x^2}+\sqrt{1-y^2}\) = a(x – y) then show that \(\frac{d y}{d x}=\sqrt{\frac{1-y^2}{1-x^2}}\). [Mar. ’17 (TS), ’08, ’05; May ’14, ’13 (Old), ’11, ’97]
Solution:
Given that \(\sqrt{1-x^2}+\sqrt{1-y^2}\) = a(x – y)
Put x = sin α ⇒ α = sin-1x
y = sin β ⇒ β = sin-1y
Now, \(\sqrt{1-\sin ^2 \alpha}+\sqrt{1-\sin ^2 \beta}\) = a(sin α – sin β)
\(\sqrt{\cos ^2 \alpha}+\sqrt{\cos ^2 \beta}\) = a(sin α – sin β)
cos α + cos β = a(sin α – sin β)
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q4
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q4.1

Question 5.
If y = \(\mathbf{x} \sqrt{\mathbf{a}^2+x^2}+a^2 \log \left(x+\sqrt{a^2+x^2}\right)\) then show that \(\frac{d y}{d x}=2 \sqrt{a^2+x^2}\). [Mar. ’19, ’15 (AP). ’09, ’02; May ’08]
Solution:
Given, that y = \(\mathbf{x} \sqrt{\mathbf{a}^2+x^2}+a^2 \log \left(x+\sqrt{a^2+x^2}\right)\)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q5
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q5.1

Question 6.
If y = \(\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)+\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)\) – \(\tan ^{-1}\left(\frac{4 x-4 x^3}{1-6 x^2+x^4}\right)\) then show that \(\frac{\mathbf{d y}}{\mathbf{d x}}=\frac{1}{1+x^2}\). [May ’07]
Solution:
Given that
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q6
y = 2θ + 3θ – 4θ = θ
y = tan-1x
Differentiating on both sides with respect to x.
\(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}} \tan ^{-1} \mathrm{x}\)
∴ \(\frac{d y}{d x}=\frac{1}{1+x^2}\)

Question 7.
If y = \(\frac{(1-2 x)^{2 / 3}(1+3 x)^{-3 / 4}}{(1-6 x)^{5 / 6}(1+7 x)^{-6 / 7}}\), then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [May ’10]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q7

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

Question 8.
If y = (sin x)log x + xsin x then find \(\frac{\mathbf{d y}}{\mathbf{d x}}\). [Mar. ’17 (AP), ’15 (TS), ’13]
Solution:
Given that, let y = (sin x)log x + xsin x
Let, u = (sin x)log x , v = xsin x then y = u + v
Differentiating on both sides with respect to x.
\(\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}\) …….(1)
Now, u = (sin x)log x
Taking logarithms on both sides
log u = log (sin x)log x
log u = log x . log (sin x)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q8
Now, v = xsin x
Taking logarithms on both sides
log v = log xsin x
log v = sin x log x
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q8.1

Question 9.
If xy + yx = ab then show that \(\frac{d y}{d x}=-\left[\frac{y x^{y-1}+y^x \log y}{x^y \log x+x y^{x-1}}\right]\). [Mar. ’03; Mar. ’16 (TS)]
Solution:
Given that, xy + yx = ab
Let, xy = u, yx = v then, u + v = ab
Differentiating on both sides with respect to x.
\(\frac{\mathrm{du}}{\mathrm{dx}}+\frac{\mathrm{dv}}{\mathrm{dx}}=0\) ……(1)
Now, u = xy
Taking logarithms on both sides
log u = log xy
log u = y log x
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q9
Now, v = yx
Taking logarithms on both sides
log v = log yx
log v = x log y
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q9.1
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q9.2

Question 10.
If f(x) = \(\sin ^{-1} \sqrt{\frac{x-\beta}{\alpha-\beta}}\) and g(x) = \(\tan ^{-1} \sqrt{\frac{x-\beta}{\alpha-x}}\) then show that f'(x) = g'(x), (β < x < α).
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q10
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q10.1
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Q10.2

Some More Maths 1B Differentiation Important Questions Long Answer Type

Question 11.
Find the derivative of 20log(tan x).
Solution:
Let y = 20log(tan x)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q1

Question 12.
If f(x) = e2x log x (x > 0), then find f'(x).
Solution:
f(x) = e2x (log x)
Let y = e2x (log x)
Differentiating with respect to ‘x’ on both sides,
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q2

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

Question 13.
If y = \(\frac{\mathbf{a}-\mathbf{x}}{\mathbf{a}+\mathbf{x}}\), find \(\frac{\mathbf{d y}}{\mathbf{d x}}\).
Solution:
Given y = \(\frac{\mathbf{a}-\mathbf{x}}{\mathbf{a}+\mathbf{x}}\)
Differentiating with respect to x on both sides.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q3
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q3.1

Question 14.
If y = sin-1(cos x) then find \(\frac{d y}{d x}\).
Solution:
Let y = sin-1(cos x)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q4

Question 15.
If x4 + y4 – a2xy = 0, find \(\frac{\mathbf{d y}}{\mathbf{d x}}\).
Solution:
Given that x4 + y4 – a2xy = 0
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q5

Question 16.
Find the derivative of cos-1(4x3 – 3x) with respect to ‘x’.
Solution:
Let y = cos-1(4x3 – 3x)
Put x = cos θ
⇒ θ = cos-1x
Now, y = cos-1(4 cos3θ – 3 cos θ)
= cos-1(cos 3θ)
= 3θ
y = 3 cos-1x
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q6

Question 17.
Find the derivative of sec x from the first principle.
Solution:
Given, f(x) = sec x
Now, f(x + h) = sec (x + h)
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q7
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q7.1
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q7.2

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

Question 18.
Find the derivative of \(\sin ^{-1}\left(\frac{b+a \sin x}{a+b \sin x}\right)\).
Solution:
Let y = \(\sin ^{-1}\left(\frac{b+a \sin x}{a+b \sin x}\right)\)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q8
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q8.1
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q8.2

Question 19.
Find the derivative of \(\tan ^{-1}\left(\frac{\cos x}{1+\cos x}\right)\)
Solution:
Let y = \(\tan ^{-1}\left(\frac{\cos x}{1+\cos x}\right)\)
Differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q9
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q9.1

Question 20.
Find the derivative of \(\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)\) with respect to tan-1x. [May ’09]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q10
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type DTP Q10.1

Question 21.
If f(x) = x ex sin x, then find f'(x).
Solution:
f(x) = x . ex . sin x
Let y = x . ex . sin x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{d}{d x}\) (x . ex . sin x)
= x . ex . \(\frac{d}{d x}\) (sin x) + x . sin x . \(\frac{d}{d x}\) (ex) + sin x . ex . \(\frac{d}{d x}\) (x)
= x . ex . cos x + x . sin x . ex + sin x . ex (1)
= ex (x cos x + x sin x + sin x)
∴ f'(x) = ex (x cos x + x sin x + sin x)

Question 22.
If f(x) = sin(log x), (x > 0), then find f'(x). [Mar. ’18 (AP)]
Solution:
f(x) = sin(log x)
Let y = sin(log x)
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q2

Question 23.
If f(x) = (x3 + 6x2 + 12x – 13)100, then find f'(x).
Solution:
Given that, f(x) = (x3 + 6x2 + 12x – 13)100
Let y = (x3 + 6x2 + 12x – 13)100
Differentiating with respect to x on both sides.
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{d}{d x}\) (x3 + 6x2 + 12x – 13)100
= 100(x3 + 6x2 + 12x – 13)100-1 \(\frac{d}{d x}\)(x3 + 6x2 + 12x – 13)
= 100(x3 + 6x2 + 12x – 13)99 (3x2 + 6(2x) + 12(1) – 0)
= 100(x3 + 6x2 + 12x – 13)99 (3x2 + 12x + 12)
= 100(x3 + 6x2 + 12x – 13)99 3(x2 + 4x + 4)
= 300(x + 2)2 (x3 + 6x2 + 12x – 13)99
∴ f'(x) = 300(x + 2)2 . (x3 + 6x2 + 12x – 13)99

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

Question 24.
Find the derivative of (ax + b)n (cx + d)m.
Solution:
Given, f(x) = (ax + b)n (cx + d)m
Let y = (ax + b)n (cx + d)m
Differentiating with respect to ‘x’ on both sides.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q4

Question 25.
Find the derivative of \(\frac{\mathbf{p} \mathbf{x}^2+\mathbf{q x}+\mathbf{r}}{\mathbf{a x}+\mathbf{b}}\).
Solution:
Given, f(x) = \(\frac{\mathbf{p} \mathbf{x}^2+\mathbf{q x}+\mathbf{r}}{\mathbf{a x}+\mathbf{b}}\)
Let y = \(\frac{\mathbf{p} \mathbf{x}^2+\mathbf{q x}+\mathbf{r}}{\mathbf{a x}+\mathbf{b}}\)
Differentiating with respect to ‘x’ on both sides.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q5

Question 26.
Find the derivative of \(\log _7(\log x)\).
Solution:
Given that, f(x) = \(\log _7(\log x)\)
Let y = \(\log _7(\log x)\)
Differentiating with respect to ‘x’ on both sides.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q6
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q6.1

Question 27.
Find the derivative of the function f(x) = (x2 – 3)(4x3 + 1). [May ’15 (AP)]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q7

Question 28.
Find the derivative of tan-1(log x). [Mar. ’19 (AP); May ’15 (TS)]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q8

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

Question 29.
If f(x) = 2x3 + 3x – 5, then prove that f'(0) + 3 . f'(-1) = 0. [Mar. ’16 (AP)]
Solution:
Given f(x) = 2x2 + 3x – 5
Now f'(x) = 2(2x) + 3(1) – 0 = 4x + 3
f'(0) = 4(0) + 3 = 3
f(-1) = 4(-1) + 3 = -4 + 3 = -1
LHS = f'(0) + 3. f'(-1)
= 3 + 3(-1)
= 3 – 3
= 0
∴ f'(0) + 3 . f'(-1) = 0

Question 30.
If 2x2 – 3xy + y2 + x + 2y – 8 = 0, then find \(\frac{d \mathbf{y}}{\mathbf{d x}}\). [Mar. ’16 (TS)]
Solution:
Given 2x2 – 3xy + y2 + 2y – 8 = 0
differentiating on both sides with respect to ‘x’.
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q10

Question 31.
If ay4 = (x + b)5 then 5yy11 = (y1)2. [Mar. ’17 (TS)]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q11
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q11.1

Question 32.
If y = x4 + tan x then find y11. [Mar. ’18 (AP)]
Solution:
Given that y = x4 + tan x
y1 = \(\frac{\mathrm{d}}{\mathrm{dx}}\)(x4 + tan x) = 4x3 + sec2x
y11 = \(\frac{\mathrm{d}}{\mathrm{dx}}\)(4x3 + sec2x)
= 4(3x2) + 2 sec x \(\frac{\mathrm{d}}{\mathrm{dx}}\)(sec x)
= 12x2 + 2 sec x (sec x tan x)
∴ y11 = 12x2 + 2 sec2x tan x

Question 33.
If y = \(\frac{2 x+3}{4 x+5}\), then find y”.
Solution:
y = \(\frac{2 x+3}{4 x+5}\)
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q13

Question 34.
If f(x) = log(tan ex), then find f'(x). [Mar. ’19 (TS)]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q14

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

Question 35.
Evaluate \({Lim}_{x \rightarrow 0} \frac{\log _e(1+5 x)}{x}\). [Mar. ’19 (TS)]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q15

Question 36.
If xlog y = log x, then show that \(\frac{d y}{d x}=\frac{y}{x}\left(\frac{1-\log x \log y}{(\log x)^2}\right)\). [Mar. ’19 (TS)]
Solution:
TS Inter First Year Maths 1B Differentiation Important Questions Long Answer Type Some More Q16

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