Inter 1st Year Maths 1B Differentiation Important Questions

Students get through Maths 1B Important Questions Inter 1st Year Maths 1B Differentiation Important Questions which are most likely to be asked in the exam.

Intermediate 1st Year Maths 1B Differentiation Important Questions

Question 1.
If f(x) = x2 (x ∈ R), prove that f is differentiable on R and find its derivative.
Solution:
Given that f(x) = x2
for x, h ∈ R, f(x + h) – f(x)(x + h)2 – x2
= x2 + h2 + 2hx – x2
= 2hx + h2 = h(2x + h)
Inter 1st Year Maths 1B Differentiation Important Questions 1
∴ f is differentiable on R and f'(x) = 2x for each x ∈ R

Inter 1st Year Maths 1B Differentiation Important Questions

Question 2.
Suppose f(x) = \(\sqrt{x}\) (x > 0). Prove that f is differentiable on (0, ∞) and find f(x).
Solution:
Let x ∈ (0, ∞) h ≠ 0 and |h| < 0
Inter 1st Year Maths 1B Differentiation Important Questions 2

Question 3.
If f(x) = \(\frac{1}{x^{2}+1}\) (x ∈ R), prove that f is differentiable on R and find f'(x).
Solution:
Let x ∈ R and h ≠ 0
Inter 1st Year Maths 1B Differentiation Important Questions 3
= –\(\frac{2 x}{\left(x^{2}+1\right)^{2}}\)
∴ f is differentiable and f'(x) = –\(\frac{2 x}{\left(x^{2}+1\right)^{2}}\) for each x ∈ R .

Question 4.
If f(x) = sin x (x ∈ R), then show that f is differentiable on R and f'(x) = cosx.
Solution:
Let x ∈ R and h ≠ 0
Inter 1st Year Maths 1B Differentiation Important Questions 4
∴ f is differentiable on R and f'(x) = cos x for each x ∈ R.

Inter 1st Year Maths 1B Differentiation Important Questions

Question 5.
Show that f(x) = |x| (x ∈ R) is not differentiable at zero and is differentiable at any x ≠ 0.
Solution:
Given f(x) = |x|
∴ f(x) = x if x ≥ 0
if h ≠ 0
\(\frac{f(0+h)-f(0)}{h}\) = \(\frac{f(h)}{h}\) = \(\left\{\begin{array}{r}
1 \text { if } h>0 \\
-1 \text { if } h<0
\end{array}\right.\)
f'(0+) = 1, f'(0) = -1
∴ is not differentiable at zero it can be easily proved that f is differentiable at any x ≠ 0 and that f'(x) = \(\left\{\begin{array}{l}
1 \quad \text { if } x>0 \\
-1 \text { if } x<0
\end{array}\right.\)

Question 6.
Check whether the following function is differentiable at zero f(x) = \(\left\{\begin{array}{l}
3+x \text { if } x \geq 0 \\
3-x \text { if } x<0
\end{array}\right.\)
Solution:
Inter 1st Year Maths 1B Differentiation Important Questions 5
f has the left hand derivative at zero and f'(0) = -1
∴ f'(0+) ≠ f'(0)
f(x) is not differentiable at zero.

Inter 1st Year Maths 1B Differentiation Important Questions

Question 7.
Show that the derivative of a constant function on an interval is zero.
Solution:
let f be a constant function on an interval I.
f(x) = C ∀ x ∈ I for some constant.
Let a ∈ I, for h ≠ 0 \(\frac{f(a+h)-f(a)}{h}\) = \(\frac{c-c}{h}\) = 0
for sufficiently small (h)
Inter 1st Year Maths 1B Differentiation Important Questions 6
∴ f is differentiable 0 and f'(0).

Question 8.
Suppose for all x, y ∈ R f(x + y) = f(x). f(y) and f'(0) exists. Then show that f(x) exists and equals to f(x) f'(0)for all x ∈ R.
Solution:
Let x ∈ R, for h ≠ 0, we have
\(\frac{f(x+h)-f(x)}{h}\) = \(\frac{f(x) f(h)-f(x)}{h}\)
= f(x) \(\frac{[f(h)-1]}{h}\) ………………….. (1)
f(0) = f(0 + 0) = f(0) f(0) ⇒ f(0) (1 —f(0)) = 0
∴ f(0) = 0 or f(0) = 1
Case (1) : Suppose f(0) = 0
f(x) = f(x + 0) = f(x) f(0) = 0 ∀ x ∈ R
∴ f(x) is a constant function = f'(x) = 0 for all x ∈ R
∴ f'(x) = 0 = f(x) . f'(0)

Case (ii): Suppose f(0) = 1
Inter 1st Year Maths 1B Differentiation Important Questions 7
∴ f is differentiable and f'(x) = f'(x) f'(0).

Question 9.
If f(x) = (ax + b)n, (x > –\(\frac{b}{a}\)), then find f'(x).
Sol:
Let u = ax + b so that y = un
f'(x) = \(\frac{\mathrm{d}}{\mathrm{dx}}\) (un) \(\frac{\mathrm{du}}{\mathrm{dx}}\)
= n.un-1a.
= an(ax+b)n – 1

Question 10.
Find the derivative of f(x) = ex (x2 + 1)
Solution:
Let u = ex, V = x2 + 1
\(\frac{\mathrm{du}}{\mathrm{dx}}\) = ex, \(\frac{\mathrm{dv}}{\mathrm{dx}}\) = 2x
f'(x) = u(x) v'(x) + u'(x) . v(x)
= ex2x + (x2 + 1) ex
= ex(2x + x2 + 1)
= ex(x + 1)2

Inter 1st Year Maths 1B Differentiation Important Questions

Question 11.
If y = \(\frac{a-x}{a+x}\) (x ≠ -a), find \(\frac{\mathrm{d} y}{\mathrm{dx}}\).
Solution:
Let u = a – x and v = a + x so that y = \(\frac{\mathrm{u}}{\mathrm{v}}\)
Inter 1st Year Maths 1B Differentiation Important Questions 8

Question 12.
If f(x) = e2x . log x (x > 0), then find f'(x).
Solution:
Let u = e2x, v = log x so that
\(\frac{\mathrm{du}}{\mathrm{dx}}\) = 2 . e2x , \(\frac{\mathrm{dv}}{\mathrm{dx}}\) = 1
f(x) = u.v
f'(x) = u . \(\frac{\mathrm{dv}}{\mathrm{dx}}\) + y . \(\frac{\mathrm{du}}{\mathrm{dx}}\)
= e2x . \(\frac{1}{x}\) + log x (2e2x)
= e2x (\(\frac{1}{x}\) + 2 logx)

Question 13.
If f(x) = \(\sqrt{\frac{1+x^{2}}{1-x^{2}}}\) (|x| < 1), then find f'(x)
Solution:
Inter 1st Year Maths 1B Differentiation Important Questions 9

Question 14.
If f(x) = x2, 2x log x (x > 0), find f'(x).
Solution:
u = x2, v = 2x, w = logx
\(\frac{\mathrm{du}}{\mathrm{dx}}\) = 2x, \(\frac{\mathrm{dv}}{\mathrm{dx}}\) = 2x . log2, \(\frac{\mathrm{dw}}{\mathrm{dx}}\) = \(\frac{1}{x}\)
f'(x) = uv . \(\frac{\mathrm{dw}}{\mathrm{dx}}\) + vw . \(\frac{\mathrm{du}}{\mathrm{dx}}\) + uw . \(\frac{\mathrm{dv}}{\mathrm{dx}}\)
= x22x . \(\frac{1}{x}\) + 2x . logx(2x) + x2 . logx . 2x log2
= x . 2x (logx2 + xlogx (log 2) + 1)

Inter 1st Year Maths 1B Differentiation Important Questions

Question 15.
If y = \(\left|\begin{array}{l}
f(x) g(x) \\
\phi(x) \psi(x)
\end{array}\right|\) then show that \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\left|\begin{array}{l}
f^{\prime}(x) g^{\prime}(x) \\
\phi(x) \psi(x)
\end{array}\right|\) + \(\left|\begin{array}{l}
f(x) g \backslash(x) \\
\phi^{\prime}(x) \Psi(x)
\end{array}\right|\)
Solution:
Given y = \(\left|\begin{array}{l}
f(x) g(x) \\
\phi(x) \psi(x)
\end{array}\right|\)
= f(x) ψ(x) – Φ(x) g(x)
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = f(x) ψ'(x) + ψ(x) f'(x) – [Φ(x). g'(x) + g(x). Φ'(x)]
= [f(x) ψ'(x) – g(x) Φ’x)] + [f'(x) ψ(x) – Φ(x).g'(x)]
= \(\left|\begin{array}{l}
f(x) g(x) \\
\phi^{\prime}(x) \psi^{\prime}(x)
\end{array}\right|\) + \(\left|\begin{array}{cc}
f^{\prime}(x) & g^{\prime}(x) \\
\phi(x) & \psi(x)
\end{array}\right|\)

Question 16.
If f(x) = 7x2+3x (x > 0), then find f'(x).
Solution:
Let u = x3 + 3x ⇒ \(\frac{\mathrm{du}}{\mathrm{dx}}\) = 3x2 + 3 = 3(x2 + 1)
f(X) = 7u
f'(x) = \(\frac{d f}{d u}\) . \(\frac{\mathrm{du}}{\mathrm{dx}}\) (7u . l0g 7) [3(x2 + 1)]
= 3(x2 + 1) 7x2+3x log 7

Question 17.
If f(x) = x ex sin x, then find f(x).
Solution:
Let u = x ,v = ex, w = sin x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 1, \(\frac{\mathrm{dv}}{\mathrm{dx}}\) = ex . \(\frac{\mathrm{dw}}{\mathrm{dx}}\) = cos x
f(x) = u.v.w
f'(x) = uv . \(\frac{\mathrm{dw}}{\mathrm{dx}}\) + uw \(\frac{\mathrm{dv}}{\mathrm{dx}}\) + vw \(\frac{\mathrm{du}}{\mathrm{dx}}\)
= xex cos x + x . sinx ex + ex sin x.

Inter 1st Year Maths 1B Differentiation Important Questions

Question 18.
If f(x) = sin (log x), (x > 0), find f'(x).
Solution:
Let u = logx, y = f(x) so that y = sin u
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{d y}{d u}\) . \(\frac{\mathrm{du}}{\mathrm{dx}}\)
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = cos u, \(\frac{\mathrm{du}}{\mathrm{dx}}\) = \(\frac{1}{x}\)
f'(x) = \(\frac{1}{x}\) . cos u = \(\frac{1}{x}\) cos (log x)

Question 19.
If f(x) =(x3 + 6x2 + 12x – 13)100; find f'(x).
Solution:
u = x3 + 6x2 + 12x – 13
⇒ \(\frac{\mathrm{du}}{\mathrm{dx}}\) = 3x2 + 12x + 12
= 3(x2 + 4x + 4)
= 3(x + 2)2
f(x) = u100
f'(x) = 100 . u99 . \(\frac{\mathrm{du}}{\mathrm{dx}}\)
= 100 (x3 + 6x2 + 12x – 13)99 . 3(x + 2)2
= 300 (x + 2)2 (x3 + 6x2 + 12x – 13)99

Question 20.
Find the derivative of f(x) = \(\frac{x \cos x}{\sqrt{1+x^{2}}}\)
Solution:
Let u = x cos x, and v = \(\sqrt{1+x^{2}}\) so that
Inter 1st Year Maths 1B Differentiation Important Questions 10

Question 21.
li f(x) = log (secx + tan x), find f'(x). [Mar 14, May 11]
Solution:
Let u = sec x + tan x and y = log u
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{1}{u}\), \(\frac{\mathrm{du}}{\mathrm{dx}}\) = sec x. tan x + sec2 x
= sec x (sec x + tan x)
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{\mathrm{dy}}{\mathrm{dx}}\) . \(\frac{\mathrm{du}}{\mathrm{dx}}\)
= \(\frac{1}{\sec x+\tan x}\) . sec x(sec x + tan x) = sec x

Inter 1st Year Maths 1B Differentiation Important Questions

Question 22.
If y = sin-1\(\sqrt{x}\), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\)
Solution:
u = \(\sqrt{x}\), y = sin-1 x.
Inter 1st Year Maths 1B Differentiation Important Questions 11

Question 23.
If y = sec (\(\sqrt{\tan x}\)), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
u = \(\sqrt{\tan x}\), v = tanx
Then y = sec u, u = \(\sqrt{\mathrm{v}}\), v = tan x
Inter 1st Year Maths 1B Differentiation Important Questions 12

Question 24.
If y = \(\frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}}\), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Let u = x sin-1x, v = \(\sqrt{1-x^{2}}\)
Inter 1st Year Maths 1B Differentiation Important Questions 13

Inter 1st Year Maths 1B Differentiation Important Questions

Question 25.
If y = log (cosh 2x), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Let u = cosh 2x, so that y = log u
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{1}{u}\); \(\frac{\mathrm{du}}{\mathrm{dx}}\) = 2 sin h2x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{d y}{d u}\) . \(\frac{\mathrm{du}}{\mathrm{dx}}\)
= 2 sin h 2x . \(\cosh 2 x\) = 2 tan h 2x

Question 26.
If y = log (sin (log x)), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Let v = log x, u = sin v so that y = log u.
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{1}{u}\); \(\frac{d u}{d v}\) = cos u; \(\frac{\mathrm{dv}}{\mathrm{dx}}\) = \(\frac{1}{x}\)
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{d y}{d u}\) . \(\frac{d u}{d v}\) . \(\frac{\mathrm{dv}}{\mathrm{dx}}\)
= \(\frac{1}{\sin (\log x)}\) . cos (logx) \(\frac{1}{x}\) = \(\frac{\cot (\log x)}{x}\)

Question 27.
If y = (cot-1x3)2, find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
u = cot-1x3, u = x3, y = u2
Inter 1st Year Maths 1B Differentiation Important Questions 14

Inter 1st Year Maths 1B Differentiation Important Questions

Question 28.
If y = cosec-1(e2x+1), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
u = e2x+1, y = cosec-1u
Inter 1st Year Maths 1B Differentiation Important Questions 15

Question 29.
If y = tan-1 (cos \(\sqrt{x}\)), find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
v = \(\sqrt{x}\) and u = cos v, y = tan-1u
\(\frac{\mathrm{dv}}{\mathrm{dx}}\) = \(\frac{1}{2 \sqrt{x}}\), \(\frac{d u}{d v}\) = – sin u; \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{1}{1+u^{2}}\)
= – sin \(\sqrt{x}\) = \(\frac{1}{1+\cos ^{2}(\sqrt{x})}\)
Inter 1st Year Maths 1B Differentiation Important Questions 16

Question 30.
If y = Tan-1 \(\sqrt{42}\) for 0 < |x| < 1, find \(\frac{\mathrm{dy}}{\mathrm{dx}}\). [May, Mar 12]
Solution:
Put x2 = cos 2θ
Inter 1st Year Maths 1B Differentiation Important Questions 17

Inter 1st Year Maths 1B Differentiation Important Questions

Question 31.
If y = x2exsin x, find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
log y = log x2. ex. sin x
= log x2 + log ex + log sin x
= 2 log x + log ex + log sin x
Differentiating w.r.to by sin x
\(\frac{1}{y}\) . \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{2}{x}\) + 1 + \(\frac{1}{sin x}\) . cos x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = y(\(\frac{2}{x}\) + 1 + cot x)

Question 32.
If y = xtanx + (sin x)cos x, find \(\frac{\mathrm{dy}}{\mathrm{dx}}\) [Mar. 14, 11]
Solution:
Let u = xtanx and v = (sin x)cos x
log u logx tanx = (tan x) log x
Inter 1st Year Maths 1B Differentiation Important Questions 18
Inter 1st Year Maths 1B Differentiation Important Questions 19

Question 33.
If x = a(cos t + log tan (\(\frac{t}{2}\))), y = a sin t, find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Inter 1st Year Maths 1B Differentiation Important Questions 20
Inter 1st Year Maths 1B Differentiation Important Questions 21

Inter 1st Year Maths 1B Differentiation Important Questions

Question 34.
If xy = ex-y, than show that \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{\log x}{(1+\log x)^{2}}\)    [May 07]
Solution:
xy = ex-y
log xy = log ex-y
y log x = (x – y) (log e = 1)
y(1 + log x) = x
Inter 1st Year Maths 1B Differentiation Important Questions 22

Question 35.
If siny = x sin (a + y), then show that \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{\sin ^{2}(a+y)}{\sin a}\) (a is not a multiple of π)
Solution:
Inter 1st Year Maths 1B Differentiation Important Questions 23

Question 36.
If y = x4 + tan x, then find y”.
Solution:
y = x4 + tan x
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 4x3 + sec2 x
\(\frac{d^{2} y}{d x^{2}}\) = 12x2 + 2 sec x (sec x tan x)
= 12x2 + 2 sec2x . tan x

Inter 1st Year Maths 1B Differentiation Important Questions

Question 37.
If f(x) = sinx, sin 2x sin 3x, find f”(x).
Solution:
f(x) = \(\frac{1}{2}\) sin 2x(2 sin 3x sin x)
= \(\frac{1}{2}\) (sin 2x) (cos 2x – cos4x)
= \(\frac{1}{4}\) (2 sin 2x cos 2x – 2 sin 2x cos 4x)
= \(\frac{1}{4}\) (sin2x + sin4x – sin6x)
Therefore,
f'(x)= \(\frac{1}{4}\)[2 cos 2x+ 4cos 4x – 6cos 6x]
Hence,
f”(x) = \(\frac{1}{4}\) (-4 sin 2x – 16 sin 4x + 36 sin 6x)
= 9 sin 6x – 4 sin 4x – sin 2x.

Question 38.
Show that y = x + tan x satisfies cos2x \(\frac{d^{2} y}{d x^{2}}\) + 2x = 2y.
Solution:
y = x + tan x implies that y’ = 1 + sec2 x
That is, y’ cos2x = 1 + cos2x.
Differentiating both sides of the above equation we get
y” cos2x + y’ . 2 cos x (-sin x) = 2 cos x (- sin x)
∴ y” cos2 x = 2(y’ – 1) sin x cos x
= 2 sec2x sin x cos x = 2 tan x = 2(y – x)
This proves the result.

Question 39.
If x = a(t – sin t),y = a(1 + cost), find \(\frac{d^{2} y}{d x^{2}}\).
Solution:
Inter 1st Year Maths 1B Differentiation Important Questions 24
Inter 1st Year Maths 1B Differentiation Important Questions 25

Inter 1st Year Maths 1B Differentiation Important Questions

Question 40.
Find the second order derivative of y = tan-1(\(\frac{2 x}{1-x^{2}}\))
Solution:
Put x = tan θ, Then
y = tan-1 (\(\frac{2 \tan \theta}{1-\tan ^{2} \theta}\))
= tan-1 (tan 2θ)
= 2θ = 2 tan-1x
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{2}{1+x^{2}}\) and \(\frac{d^{2} y}{d x^{2}}\) = \(\frac{-4 x}{\left(1+x^{2}\right)^{2}}\).

Question 41.
If y = sin (sin x), show that y” + (tan x) y’ + y cos2x = 0.
Solution:
y = sin (sin x) implies that .
y’ = cos x . cos (sin x) and
y” = -cos2x sin (sin x) – sin x cos (sin x)
= – y cos2x – sin x (\(\frac{y^{\prime}}{\cos x}\))
= -y cos2x – y’ tan x
∴ y” + (tan x)y’ + y cos2 x = 0.

Question 42.
If f(x) = ex(x ∈ R), then show that f(x) = ex by first principle.
Solution:
From f(x) = ex we have for h ≠ 0
Inter 1st Year Maths 1B Differentiation Important Questions 26

Inter 1st Year Maths 1B Differentiation Important Questions

Question 43.
If f(x) = log x (x > 0), then show that f(x) = \(\frac{1}{x}\) by first principle.
Solution:
Now for h ≠ 0
Inter 1st Year Maths 1B Differentiation Important Questions 27
\(\frac{d}{dx}\) (log x) = \(\frac{1}{x}\)

Question 44.
If 1(x) = ax (x ∈ R) (a > 0), then show that f'(x) = ax log a by first principle.
Solution:
For h ≠ 0
\(\frac{f(x+h)-f(x)}{h}\) = \(\frac{a^{x+h}-a^{x}}{h}\) = ax [latex]\frac{a^{h}-1}{h}[/latex]
We know that \(\frac{a^{h}-1}{h}\) → log a as h → 0
Hence f'(x) = ax . log a.
\(\frac{d}{d x}\) = (ax) = ax log a

Question 45.
If y = Tan-1 \(\sqrt{\frac{1-x}{1+x}}\) (|x| < 1), we shall find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Substituting x cos u (u ∈ (0, π)) in y, we get
Inter 1st Year Maths 1B Differentiation Important Questions 28
Inter 1st Year Maths 1B Differentiation Important Questions 29
observe that Tan-1x, \(\sqrt{\frac{1-x}{1+x}}\) and cos u are the functions that stand for f(x), g(x) and h(u) respectively, mentioned in the method.

Inter 1st Year Maths 1B Differentiation Important Questions

Question 46.
If y = Tan-1[latex]\frac{2 x}{1-x^{2}}[/latex] (|x| < 1) then we shall \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Substituting x = tan u
Inter 1st Year Maths 1B Differentiation Important Questions 30

Question 47.
If x = a cos3t, y = a sin3t, find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Here \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 3a cos2t (-sin t) and
\(\frac{d y}{d t}\) = 3a sin2t. cost.
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}\) = -tan t

Question 48.
If y = et +cost, x = log t + sin t find \(\frac{\mathrm{dy}}{\mathrm{dx}}\)
Solution:
Here \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = et – sin t and \(\frac{d x}{d t}\) = \(\frac{1}{t}\) + cos t
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{t\left(e^{t}-\sin t\right)}{(1+t \cos t)}\)

Inter 1st Year Maths 1B Differentiation Important Questions

Question 49.
To find the derivative of f(x) = x\(\sin ^{\frac{1}{x}}\) with respect to g(x) = sin-1x, we have to compute \(\frac{d f}{d g}\)
Solution:
Now f(x) = x\(\sin ^{\frac{1}{x}}\) implies that
log f(x) = sin-1x . log x so that
Inter 1st Year Maths 1B Differentiation Important Questions 31

Question 50.
If x3 + y3 – 3axy = 0, find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Let the given equation define the function.
y = 1(x) that is x3 + (f(x))3 – 3axf(x) = 0
Differentiating both sides of this equation with respect to x, we get
3x2 + 3 (f(x))2 f'(x) – [3a. f(x) + 3axf'(x)] = 0
Hence 3x2 + 3y2 f'(x) – [3ay + 3ax f'(x)] = 0
∴ f'(x) = \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = \(\frac{a y-x^{2}}{y^{2}-a x}\)

Question 51.
If 2x2 – 3xy + y2 + x + 2y – 8 = 0, find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Treating y as a function of x and then differentiating with respect to x,
we get 4x – 3y – 3xy’ + 2yy’ + 1 + 2y’ = 0
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = y’ = \(\frac{3 y-4 x-1}{2 y-3 x+2}\)

Inter 1st Year Maths 1B Differentiation Important Questions

Question 52.
If y = xx (x > 0), we shall find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Taking logarithms on both the sides of
y = xx we obtain log y = x log x
Differentiating with respect to x,
We get \(\frac{y^{\prime}}{y}\) = x . \(\frac{1}{x}\) + log x = 1 + log x
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = y’ = y(1 + log x) = xx (1 + log x)

Question 53.
If y = (tan x)sin x [o < x < \(\frac{\pi}{2}\) ] compute \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
Solution:
Taking logarthms on both sides of
y= (tan x)sin x, we get
log y = sin x . log (tan x)
Differentiating with respect to x, we get
\(\frac{y^{\prime}}{y}\) = \(\frac{\sin x}{\tan x}\) . sec2x + cosx . log (tan x)
= sec x + cos x . log (tan x)
Hence \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = (tan x)sin x [sec x + cos x log (tan x)]