OP Malhotra Class 12 Maths Solutions Chapter 8 Differentiation Ex 8(e)

Effective S Chand ISC Maths Class 12 Solutions Chapter 8 Differentiation Ex 8(e) can help bridge the gap between theory and application.

S Chand Class 12 ICSE Maths Solutions Chapter 8 Differentiation Ex 8(e)

Differentiate w.r.t. x :

Question 1.
(i) e^3x
(ii) e^{cos x}
(iii) e^-x/2
(iv) \(e^{x^2+2 x}\)
(v) \(e^{\sqrt{1+x+x^2}}\)
(vi) \(e^{\sin \sqrt{x}}\)
(vii) \(e^{\frac{x^2}{1+x^2}}\)
Solution:
(i) Let y = e^3x
∴ \(\frac{d y}{d x}=e^{3 x} \frac{d}{d x} 3 x\) = [∵ \(\frac { d }{ dx }\)e^x = e^x]
= 3e^3x

(ii) Let y = e^{cos x};
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=e^{\cos x} \frac{d}{d x}(\cos x)\) = – sin x e^{cos x}

(iii) Let y = e^-x/2
Diff both sides w.r.t. x ; we get
∴ \(\frac{d y}{d x}=e^{-x / 2} \frac{d}{d x}\left(\frac{-x}{2}\right)=\frac{-1}{2} e^{-x / 2}\)

(iv) Let y = \(e^{x^2+2 x}\) ;
Diff both sides w.r.t. x ; we get
∴ \(\frac{d y}{d x}=e^{x^2+2 x} \frac{d}{d x}\left(x^2+2 x\right)\)
= 2(x + 1)e^x²+2x

(v) Let y = \(e^{\sqrt{1+x+x^2}}\)
Diff both sides w.r.t. x ; we get
∴ \(\frac{d y}{d x}=e^{\sqrt{1+x+x^2}} \frac{d}{d x} \sqrt{1+x+x^2}\)
= \(e^{\sqrt{1+x+x^2}} \frac{1}{2}\left(1+x+x^2\right)^{\frac{1}{2}-1} \frac{d}{d x}\left(1+x+x^2\right)\)
= \(e^{\sqrt{1+x+x^2}} \frac{1}{2 \sqrt{1+x+x^2}}(1+2 x)\)

(vi) \(e^{\sin \sqrt{x}}\);

Question 2.
(i) 3^x
(ii) 8^{cos x}
(iii) a^{sin x}
(iv) a^3x²
(v) 5^{log sin x}
(vi) 10^10x
(vii) \(2^{\frac{x}{\log x}}\)
Solution:
(i) Let y = 3^x
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x} = 3^x \log 3\) [∵ \(\frac { d }{ dx }\) = a^x log a]

(ii) Let y = 8^{cos x}
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=8^{\cos x} \log 8 \frac{d}{d x} \cos x\)
= – sin x 8^{cos x} log 8

(iii) Let y = a^{sin x}
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=a^{\sin x} \log a \frac{d}{d x} \sin x\)
= a^{sin x} log a. cos x

(iv) Let y = a^3x²
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=a^{3 x^2} \log a \cdot 6 x\)

(v) Let y = \(5^{\log \sin x}\)

Question 3.
(i) x e^-x
(ii) e^x cot x
(iii) e^ax bx
(iv) \(\frac{e^x}{x}\)
(v) \(\frac{e^x}{1+\sin x}\)
(vi) \(x e^{x^2}\)
(vii) \(e^{x \sin x}\)
(viii) e^ax cos (bx + c)
(ix) x² e^x sin x
(x) sin (e^x log x)
(xi) e^ax cos (b tan x)
(xii) \(e^{x^2} \log _{10}(2 x)\)
Solution:
(i) Let y = x e^-x
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=x \frac{d}{d x} e^{-x}+e^{-x} \frac{d}{d x} x\)
= x e^-x (-1) + e^-x.1 = (1 – x)e^-x

(ii) Let y = e^x cot x
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=e^x \frac{d}{d x} \cot x+\cot x \frac{d}{d x} e^x\)
= e^x (-cosec²x) + cot x e^x
= e^x [cot x – cosec²x]

(iii) Let y = e^ax sin bx
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x} = e^{a x} \frac{d}{d x} \sin b x+\sin b x \frac{d}{d x} e^{a x}\)
= b e^ax cos bx + a sin bx e^ax = em (b cos bx + a sin bx)

(iv) Let y = \(\frac{e^x}{x}\);
Diff both sides w.r.t. x ; we get

(v) Let y = \(\frac{e^x}{1+\sin x}\)
Diff both sides w.r.t. x ; we get

(vi) Let y = \(x e^{x^2}\) ;
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=x \frac{d}{d x} e^{x^2}+e^{x^2} \frac{d}{d x}(x)=x \cdot e^{x^2} \cdot 2 x+e^{x^2} \cdot 1=e^{x^2}\left(2 x^2+1\right)\)

(vii) Let y = e^x sin x
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=e^{x \sin x} \frac{d}{d x}(x \sin x)=e^{x \sin x}[x \cdot \cos x+\sin x]\)
= e^{x sinx} (xcosx + sinx)

(viii) Let y = e^ax cos (bx + c)
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=e^{a x} \frac{d}{d x} \cos (b x+c)+\cos (b x+c) \frac{d}{d x} e^{a x}\)
= e^ax {-sin(bx + c)}b + cos(bx + c)e^ax.a
= e^ax [a cos(bx + c) – b sin(bx + c)]

(ix) Let y = x² e^x sin x
Diff both sides w.r.t. x ; we get

(x) Let y = sin (e^x log x) ;
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=\cos \left(e^x \log x\right) \frac{d}{d x} e^x \log x\)
= \(\cos \left(e^x \log x\right)\left[e^x \cdot \frac{1}{x}+\log x \cdot e^x\right]\)
= \(e^x \cos \left(e^x \log x\right)\left(\frac{1}{x}+\log x\right]\)

(xi) Let y = e^ax cos (b tan x)
Diff both sides w.r.t. x ; we get

(xii) Let y = \(e^{x^2}\) log₁₀(2x);
Diff both sides w.r.t. x ; we get

Question 4.
(i) log (e^x + e^-x)
(ii) \(\log \left(\frac{e^x}{e^x+1}\right)\)
(iii) \(\frac{e^x-e^{-x}}{e^x+e^{-x}}\)
(iv) \(\log x+e^{\sqrt{x}}\)
(v) \(e^{\tan x} \times \log \tan x\)
(vi) \(\frac{e^x+\log x}{\sin 3 x}\)
Solution:
(i) Let y = log (e^x + e^-x) ;
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=\frac{1}{e^x+e^{-x}} \frac{d}{d x}\left(e^x+e^{-x}\right)\)
\(\frac{e^x-e^{-x}}{e^x+e^{-x}}\)

(ii) Let y = log\(\left(\frac{e^x}{e^x+1}\right)\)
= log e^x – log (e^x + 1)
⇒ y = x log e – log (e^x + 1)
⇒ y = x – log (e^x + 1) ;
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=1-\frac{1}{e^x+1} e^x\)
\(\frac{e^x+1-e^x}{e^x+1}=\frac{1}{e^x+1}\)

(iii) Let y = \(\frac{e^x-e^{-x}}{e^x+e^{-x}}\)
Diff both sides w.r.t. x ; we get

Question 5.
\(e^{\log \left(x+\sqrt{x^2-a^2}\right)}\)
Solution:

Question 6.
\(e^{-a x^2} \sin (\log x)\)
Solution:

Question 7.
\(\log x \cdot e^{\left(\tan x+x^2\right)}\)
Solution:

Question 8.
e^x log sin 2x
Solution:
Let y = e^x log sin 2x ;
Diff both sides w.r.t. x ; we get
\(\frac{d y}{d x}=e^x \frac{1}{\sin 2 x} \frac{d}{d x} \sin 2 x+\log \sin 2 x \frac{d}{d x} e^x\)
\(e^x \frac{1}{\sin 2 x} \cos 2 x \cdot 2+\log \sin 2 x \cdot e^x=e^x(2 \cot 2 x+\log \sin 2 x)\)

Question 9.
\(e^{\sin x} \cdot \sin e^x\)
Solution:

Question 10.
e^x log(1 + x²)
Solution: