KC Sinha Mathematics Solution Class 12 Chapter 12 द्वितीय कोटि का अवकलज (Second Order Derivative) Exercise 12.1 (Q1-Q3)

Exercise 12.1

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Question 1

निम्नलिखित फलनों के द्वितीय(कोटि) का अवकलज ज्ञात करें ।
[Find the second order derivatives of the following functions]

(i) log x
Sol :
माना y=log x

Differentiating w.r.t x

$\frac{d y}{dx}=\frac{1}{x}$

Again, Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=-\frac{1}{x^{2}}$


(ii) x²⁰
Sol :
माना y=x²⁰

Differentiating w.r.t x

$\frac{d y}{dx}=20 x^{19}$

Again , differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=380 x^{18}$

(iii) log(log x)
Sol :
माना y=log(log x)

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{1}{\log x} \times \frac{1}{x}$

$\frac{d y}{d x}=\frac{1}{x \log x}$

Again , Differentiating w.r.t x

$\frac{d^{2}y}{d x^{2}}=\frac{0 \cdot x \log x-1 \cdot\left[1 \cdot \log {x}+x \times \frac{1}{x}\right]}{(x \log x)^{2}}$

$=\frac{-\left(\log x+1\right)}{\left(x \log x\right)^{2}}$


(iv) x²+3x+2
Sol :
माना y=x²+3x+2

Differentiating w.r.t x

$\frac{d y}{d x}=2 x+3$

Again, differentiating w.r.t x

$\frac{d^{2} y}{dx^{2}}=2$


(v) xcosx
Sol :
माना y=xcosx

Differentiating w.r.t x

$\frac{d y}{d x}=1 \cdot \cos x+x(-\sin x)$

$\frac{d y}{d x}=\cos x-x \sin x$

Again, Differentiate w.r.t x

$\frac{d^{2} y}{d x^{2}}=-\sin x-[1 \cdot \operatorname{sin} x+x \cos x]$

=-sinx-sinx-xcosx

=-xcosx-2sinx


(vi) e^xsin5x
Sol :
माना y=e^xsin5x

Differentiating w.r.t x

$\frac{d y}{dx}=e^{x} \cdot \sin 5 x+e^{x} \cdot \cos 5x \times 5$

$\frac{d y}{d x}=e^{x} \sin 5 x+5e ^{2} \cos x$

Again, Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=e^{x} \sin 5 x+e^{x} \cos 5 x \times 5+\left[e^{x} \cos 5 x+e^{2}(-\sin 5x) \times 5\right]$

=e^xsin5x+5e^xcos5x+5e^xcos5x-25e^xsin5x

=10e^xcos5x-24e^xsin5x

=2e^x(5cos5x-12sin5x)


(vii) sin(log x)
Sol :
माना y=sin(log x)

Differentiating w.r.t x

$\frac{d y}{dx}=\cos (\log x) \times \frac{1}{x}$

$\frac{d y}{dx}=\frac{\cos (\log x)}{x}$

Again, Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=\frac{-\sin (\log x) \times \frac{1}{x} \times x-\cos (\log x) \times 1}{x^{2}}$

$=-\frac{\sin\left(\log x\right)+\cos \left(\log x\right)}{x^{2}}$

Question 2

यदि(If) y=sin(log x) , सिद्ध करे कि (prove that) $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0$
Sol :
y=sin(log x)

Differentiating w.r.t x

$\frac{d y}{d x}=\cos (\log x) \times \frac{1}{x}$

$\frac{dy}{dx}=\frac{\cos\left(\log x\right)}{x}$

Again , Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=\frac{-\sin \left(\log x\right) \times \frac{1}{x}.x-\cos \left(\log x\right) \cdot 1}{x^{2}}$

$\frac{d^{2} y}{d x^{2}}=-\frac{[\sin 4(\log x)+\cos (\log x)]}{x^{2}}$

$x^{2} \frac{d^{2} y}{d x^{2}}=-\sin (\log x)-\cos (\log x)$

$x^{2} \frac{d^{2} y}{d x^{2}}=-y-x \cdot \frac{d y}{dx}$

$x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{dx}+y=0$

Question 3

यदि(If) y=x³+tanx  , दिखाएँ कि(show that) $\frac{d^{2} y}{d x^{2}}=6 x+2 \sec ^{2} x \tan x$
Sol :
y=x³+tanx

Differentiating w.r.t x

$\frac{d y}{d x}=3 x^{2}+\sec ^{2} x$

Again ,Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=6 x+2 \sec x \cdot \sec x \tan x$

$=6 x+2 \operatorname{sec}^{2} x \tan x$