KC Sinha Solution Class 11 Chapter 26 अवकलज (Derivatives) Exercise 26.3

 Exercise 26.3

Question 1

x के सापेक्ष निम्नलिखित का अवकल गुणांक निकाले।

[Find the derivatives (differential coefficient) of the following functions w.r.t x]

(i) 10x5

Sol :

माना y=10x5

Differentiating w.r.t x

x के सापेक्ष अवकलन करने पर

$\frac{d y}{d x}=\frac{d\left(10 x^{5}\right)}{d x}$

$=\frac{10 d\left(x^{5}\right)}{d x}$

=10×5x4

=50x4


(ii) $2 x-\frac{3}{4}$
Sol :
माना $y=2 x-\frac{3}{4}$

Differentiating w.r.t x

x के सापेक्ष अवकलन करने पर

$\frac{d y}{dx}=\frac{d(2 a)}{d x}-\frac{d\left(\frac{3}{4}\right)}{dx}$

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


(iii) $4 \sqrt{x}-2$
Sol :
माना $y=4 \sqrt{x}-2$

Differentiating w.r.t x

x के सापेक्ष अवकलन करने पर

$\frac{d y}{d x}=\frac{d(4 \sqrt{x})}{d x}-\frac{d(2)}{dx}$

$=4\times {\frac{1}{2 \sqrt{x}}}-0$

$=\frac{2}{\sqrt{x}}$


(iv) $\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$
Sol :
माना $y=\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$

y=ax-4-bx-2+cos x

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{a d\left(x^{-4}\right)}{d x}-\frac{b \cdot d\left(x^{-2}\right)}{d x}+\frac{d(\cos x)}{d x}$

y=a(-4)x-5-b(-2)x-3-sin x

$=\frac{-4 a}{x^{5}}+\frac{2 b}{x^{3}}-\operatorname{sin} x$


(iv) $\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$
Sol :
माना y=3cosec x

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{3 \cdot d(\cos e x)}{d x}$

$=-3 \operatorname{cosec x} \cot x$


(vi) $\sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}$ जहाँ(where)$\pi<x<\frac{3 \pi}{2}$
Sol :
माना $y=\sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}$

$y=\sqrt{\frac{2 \cos ^{2} x}{2\sin ^2 x}}$

$y=\sqrt{\cot ^{2} x}$

y=cotx

Differentiating w.r.t x

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


(vii) $2\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}$
Sol :
माना $y=2\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}$

y=2(1+sin x)
y=2+2sin x

Differentiating w.r.t. x

$\frac{d y}{d x}=d \frac{(2)}{dx}+2 \frac{d(\sin x)}{dx}$

=2cos x


Question 2

यदि(If) $y=\frac{1}{1+x^{n-m}+x^{p-m}}+\frac{1}{1+x^{m-n}+x^{p-n}}+\frac{1}{1+x^{m-p}+x^{n-p}}$ ,.साबित करे कि (Show that) $\frac{d y}{d x}=0$

Sol :
$y=\frac{1}{1+x^{n-m}+x^{p-m}}+\frac{1}{1+x^{m-n}+x^{p-n}}+\frac{1}{1+x^{m-p}+x^{n-p}}$

$y=\frac{1}{1+\frac{x^{n}}{x^{m}}+\frac{x^{p}}{x^{m}}}+\frac{1}{1+\frac{x^{m}}{x^{n}}+\frac{x^{p}}{x^{n}}}+\frac{1}{1+\frac{x^{m}}{x^{p}}+\frac{x^{n}}{x^{p}}}$

$y=\frac{1}{\frac{x^{m}+x^{n}+x^{p}}{x^{m}}}+\frac{1}{\frac{x^{n}+x^{m}+x^{p}}{x^{n}}}+\frac{1}{\frac{x^{p}+x^{m}+x^{n}}{x^p}}$

$y=\frac{x^{m}}{x^{m}+x^{n}+x^{p}}+\frac{a^{n}}{x^{m}+x^{n}+x^{p}}+\frac{x^{p}}{x^{m}+x^{n}+x^{p}}$

$y=\frac{x^{m}+x^{n}+x^{p}}{x^{m}+x^{n}+x^{p}}$

y=1

Differentiating w.r.t x

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

$\frac{d y}{d x}=0$


Question 3

यदि(If) f(x)=xतथा (and) '(1)=5 ,साबित करे कि(show that) n=5
Sol :
f(x)=xn

Differentiating w.r.t x

$f^{\prime}(x)=n \cdot x^{n-1}$

x=1,

$f^{\prime}(1)=n(1)^{n-1}$

5=n×1

5=n

Question 4

x के सापेक्ष निम्नलिखित फलनो का अवकल गुणांक निकाले ।
[Find the derivatives(differential coefficient) of the following functions w.r.t x]
(i) 3x5+3x3-5
Sol :
माना y=3x5+3x3-5

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{3 \cdot d\left(x^{5}\right)}{d x}+3 \cdot \frac{d\left(x^{3}\right)}{d x}-\frac{d(5)}{dx}$

=3×5x4+3×x2-0
=15x4+9x2

(ii) 3sinx+2sinα, जहाँ α एक अचर है(where α is a constant)
Sol :
माना y=3sinx+2sinα

Differentiating w.r.t. x

$\frac{d y}{d x}=\frac{3 \cdot d\left(\sin x\right)}{dx}+\frac{d(2\sin \alpha)}{d x}$

=3cosx+0
=3cos x 


(iii) (x-2)(x-3)
Sol :
y=(x-2)(x-3)
y=x2-5x+6

Differentiating w.r.t x

$=\frac{d y}{d x}=\frac{d\left(x^{2}\right)}{d x}-5 \cdot \frac{d(x)}{d x}+\frac{d(6)}{d x}$

=2x-5+0
=2x-5


(iv) cotx+2cosx+3sinx
Sol :
माना y=cotx+2cosx+3sinx

Differentiating w.r.t x

$\frac{dy}{dx}=\frac{d(\cot x)}{dx}+2 \cdot \frac{d(\cos x)}{d x}+3 d \frac{\sin x}{d x}$

=-cosecx2-2sinx+3cosx


(v) $\sqrt{x}-\frac{1}{\sqrt{x}}$
Sol :
माना $y=\sqrt{x}-\frac{1}{\sqrt{x}}$

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

$y=\sqrt{x}-x^{-1 / 2}$

Differentiating w.r.t. x

$\frac{d y}{dx}=\frac{d(\sqrt{x})}{dx}-\frac{d\left(x^{-\frac{1}{2}}\right)}{dx}$

$=\frac{1}{2 \sqrt{x}}-\left(\frac{-1}{2}\right) x^{-3 / 2}$

$=\frac{1}{2 \sqrt{x}}+\frac{1}{2 x^{3 / 2}}$

$=\frac{1}{2 \sqrt{x}}+\frac{1}{2 x \cdot \sqrt{x}}$

$=\frac{x+1}{2 x \sqrt{x}}$

$=\frac{x+1}{2 x^{3 / 2}}$


(vi) cosx-2sinx
Sol :
माना y=cosx-2sinx

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{d(\cos x)}{d x}-2 \cdot \frac{d(\operatorname{sin} x)}{d x}$

=-sinx-2cosx

(vii) x3+7x2+4x+9
Sol :
माना y=x3+7x2+4x+9

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{d\left(x^{3}\right)}{d x}+7 \cdot \frac{d\left(x^{2}\right)}{d x}+4 \cdot \frac{d(x)}{d x}+\frac{d(9)}{d x}$

=3x2+7×2x+4(1)+0
=3x2+14x+4

(viii) 6x100-x55+x
Sol :
माना y=6x100-x55+x

Differentiating w.r.t x

$\frac{d y}{dx}=6 \cdot \frac{d\left(x^{100}\right)}{dx}-\frac{d\left(x^{55}\right)}{d x}+\frac{d(x)}{d x}$

=6×100x99-55x54+1
=600x99-55x54+1


(ix) 3cotx+5cosecx
Sol :
माना y=3cotx+5cosecx

Differentiating w.r.t x

$\frac{d y}{dx}=\frac{3 \cdot d(\cos x)}{d x}+5 \cdot d \frac{(\cos x)}{d x}$

=-3cosec2x-5cosecx cotx

(x) 5secx+4cosx
Sol :
माना y=5secx+4cosx

Differentiating w.r.t x

$\frac{d y}{dx}=\frac{5d(\tan x)}{d x}+4 \cdot \frac{d(\cos x)}{d x}$

=5secxtanx-4sinx


(xi) 2tanx-7secx
Sol :
माना y=2tanx-7secx

Differentiating w.r.t x

$\frac{d y}{dx}=\frac{2 \cdot d(\tan x)}{dx}-\frac{7d(\operatorname{sec} x)}{d x}$

=2sec2x-7secxtanx


(xii) 5sinx-6cosx+7
Sol :
माना y=5sinx-6cosx+7

Differentiating w.r.t x

Question 5

निम्नलिखित फलनो का x के सापेक्ष अवकलन करे।
[Differentiate the following functions w.r.t. to x]
(i) ax3+bx2+cx+d
Sol :
माना y=ax3+bx2+cx+d

Differentiating w.r.t x
x के सापेक्ष अवकलन करने पर

$\frac{d y}{d x}=\frac{a \cdot d\left(x^{3}\right)}{d x}+b \cdot \frac{d\left(x^{2}\right)}{d x}+c \frac{d(x)}{dx}+\frac{d(d)}{d x}$

=a3x2+b2x+c×1+0
=3ax2+2bx+c

(ii) x4+7x3+8x2+3x+2
Sol :
माना y=x4+7x3+8x2+3x+2

Differentiating w.r.t x
x के सापेक्ष अवकलन करने पर

$\frac{d y}{dx}=\frac{d\left(x^{4}\right)}{d x}+7 \cdot \frac{d\left(x^{3}\right)}{d x}+8 \cdot \frac{d\left(x^{2}\right)}{d x}+3 d \frac{(x)}{d x}+\dfrac{d(2)}{dx}$

=4x3+7×3x2+8×2x+3×1+0
=4x3+21x2+16x+3


(iii) x3+sinx
Sol :
माना y=x3+sinx

Differentiating w.r.t x
x के सापेक्ष अवकलन करने पर

$\frac{d y}{d x}=\frac{d\left(x^{3}\right)}{d x}+\frac{d(\sin x)}{dx}$

=3x2+cosx


(iv) x3+7x+3+4x2n+5a2,जहाँ a एक अचर है(where a is constant)
Sol :
माना y=x3+7x+3+4x2n+5a2

Differentiating w.r.t x
x के सापेक्ष अवकलन करने पर

=3x2+7+0
=3x2+7

(v) xn+axn-1+a2xn-2+..+an-1x+an, जहाँ a एक अचर वास्तविक संख्या है(where a is a fixed real number)
Sol :
y=xn+axn-1+a2xn-2+..+an-1x+an

Differentiating w.r.t x
x के सापेक्ष अवकलन करने पर

$\frac{d y}{dx}=\frac{d\left(x^{n}\right)}{d x}+a \cdot \frac{d\left(x^{n-1}\right)}{d x}+a^{2} \cdot \frac{d\left(x^{n-z}\right)}{d x}+\ldots+a^{n-1} \cdot \frac{d(x)}{dx}+\frac{d(a^n)}{dx}$

=nxn-1+a(n-1)xn-2+a2(n-2)xn-3+..+an-1


(vi) anxn+an-1xn-1+...+a1x+a0, जहाँ सभी a1 एक अचर है तथा an≠0 (where a1 are constant and an≠0)
Sol :
माना y=anxn+an-1xn-1+...+a1x+a0

Differentiating w.r.t x
x के सापेक्ष अवकलन करने पर

$\frac{d y}{dx}=a_{n} \cdot \frac{d\left(x^{n}\right)}{dx}+a_{n-1} \cdot \frac{d\left(x^{4-1}\right)}{dx}+\cdots \cdot \cdot a_1 \frac{d(x)}{dx}+\frac{d(a_0)}{dx}$

=annxn-1+an-1(n-1)xn-2+...+a1


Question 6

निम्नलिखित फलनो का x के सापेक्ष अवकलन करे।
[Find the d.c. of the following functions w.r.t x]

(i) $\frac{5 x^{4}+6 x^{2}-x+1}{x}$
Sol :
माना $y=\frac{5 x^{4}+6 x^{2}-x+1}{x}$

$y=\frac{5 x^{4}}{x}+\frac{6 x^{2}}{x}-\frac{x^{1}}{x}+\frac{1}{x}$

$y=5 x^{3}+6 x-1+\frac{1}{x}$

Differentiating w.r.t x

$\frac{d y}{d x}=5 \cdot \frac{d\left(x^{3}\right)}{d x}+6 \cdot \frac{d(x)}{d x}-\frac{d(1)}{dx}+\frac{d\left(\frac{1}{x}\right)}{dx}$

$=5 \times 3 x^{2}+6 \times 1-0-\frac{1}{x^{2}}$

$=15 x^{2}+6-\frac{1}{x^{2}}$
 

(ii) $\frac{(x+5)\left(2 x^{2}-1\right)}{x}$
Sol :
माना $y=\frac{(x+5)\left(2 x^{2}-1\right)}{x}$

$y=\frac{2 x^{3}-x+10 x^{2}-5}{x}$

$y=\frac{2 x^{3}}{x}-\frac{x}{x}+\frac{10 x^{2}}{x}-\frac{5}{x}$

$y=2 x^{2}-1+10 x-\frac{5}{2}$

Differentiating w.r.t x

$\frac{d y}{dx}=2 \cdot \frac{d\left(x^{2}\right)}{dx}-\frac{d(1)}{dx}+10 \cdot \frac{d(x)}{dx}-5 \cdot \frac{d\left(\frac{1}{x}\right)}{d x}$

$=2 \times 2 x-0+10-5\left(\frac{-1}{x^{2}}\right)$

$=4 x+10+\frac{5}{x^{2}}$


(iii) $\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}$
sol :
$y=\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}$

$y=(\sqrt{x})^{2}+\left(\frac{1}{\sqrt{x}}\right)^{2}+2 \cdot \sqrt{x} \cdot \frac{1}{\sqrt{x}}$

$y=x+\frac{1}{x}+2$

Differentiating w.r.t x

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


(iv) $\frac{x^{2} \tan x+1}{\tan x}$
Sol :
$y=\frac{x^{2} \tan x+1}{\tan x}$

$=\frac{x^{2} \tan x}{\tan x}+\frac{1}{\tan x}$

y=x2+cotx

Differentiating w.r.t x

$\frac{d y}{dx}=\frac{d\left(x^{2}\right)}{dx}+d \frac{(\cot x)}{d x}$

=2x-cosec2x


(v) (x-a)(x-b), जहाँ a और b अचर है(where a and b are constant)
Sol :
माना y=(x-a)(x-b)

y=x2-bx-ax+ab

Differentiating w.r.t x

$\frac{d y}{dx}=\frac{d\left(x^{2}\right)}{d x}-b \cdot \frac{d(x)}{dx}-a \cdot \frac{d(x)}{dx}+\frac{d(a s)}{dx}$

=2x-b-a+0
=2x-a-b

(vi) (ax2+b)2,जहाँ a और b अचर है(where a and b are constant)
Sol :
माना y=(ax2+b)2

y=(ax2)2+2.ax2+b+b2

y=a2x4+2abx2+b2

Differentiating w.r.t x

$\frac{d y}{d x}=a^{2} \cdot d \frac{\left(x^{4}\right)}{dx}+\frac{2 a b d\left(x^{2}\right)}{dx}+\frac{d\left(b^{2}\right)}{dx}$

=a24x3+2ab×2x+0
=4ax(ax2+b)


Question 7

$\frac{d y}{d x}$ निकाले(Find $\frac{d y}{d x}$ when)
(i) $y=8 x^{2}-x+5-\frac{3}{x^{3}}$
Sol :
Differentiating w.r.t x

$\frac{d y}{dx}=8 \cdot \frac{d\left(x^{2}\right)}{d x}-\frac{d(x)}{dx}+\frac{d(5)}{d}-3 d \frac{\left(x^{-3}\right)}{dx}$

=8×2x-1+0-3(-3)x-4

=16x-1$+\frac{9}{x^4}$


(ii) $y=\frac{2 x^{2}-3 x+1}{\sqrt{x}}$
Sol :
$y=\frac{2 x^{2}-3 x+1}{x^{1 / 2}}$

$y=\frac{2 x^{2}}{x^{1 / 2}}-\frac{3 x}{x^{\frac{1}{2}}}+\frac{1}{x^{1 / 2}}$

$y=2 x^{3 / 2}-3 x^{1 / 2}+x^{-\frac{1}{2}}$

Differentiating w.r.t x

$\frac{dy}{dx}=2 \frac{d\left(x^{3 / 2}\right)}{dx}-3 \frac{d\left(x^{\frac{1}{2}}\right)}{dx}+d \frac{\left(x^{-\frac{1}{2}}\right)}{dx}$

$=2 \cdot\left(\frac{3}{2}\right) x^{\frac{1}{2}}-3\left(\frac{1}{2}\right) x^{-1 / 2}+\left(\frac{-1}{2}\right)x^{-3 / 2}$

$=\frac{6}{2} x^{\frac{1}{2}} \cdot \frac{-3}{2 x^{1 / 2}}-\frac{1}{2 x^{3 / 2}}$

$=\frac{6 x^{2}-3 x-1}{2 x^{3 / 2}}$


(iii) $y=\frac{x^{3}+5 x^{2}+x+7}{x}$
Sol :
$y=\left(x^{2}\right)^{3}+3 \cdot\left(x^{2}\right)^{2} \cdot \frac{1}{x^{2}}+3 \cdot x^{2} \cdot\left(\frac{1}{x^{2}}\right)^{2}+\left(\frac{1}{x^{2}}\right)^{3}$

$y=x^{6}+3 x^{2}+\frac{3}{x^{2}}+\frac{1}{x^{6}}$

$y=x^{6}+3 x^{2}+3 x^{-2}+x^{-6}$

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{d\left(x^{6}\right)}{dx}+\frac{3 \cdot d\left(x^{2}\right)}{dx}+3 \cdot \frac{d\left(x^{-2}\right)}{dx}+\frac{d\left(x^{-6}\right)}{d x}$

=6x5+3×2x+3(-2)x-3+(-6)x-7

$=6 x^{5}+6 x-\frac{6}{x^{3}}-\frac{6}{x^{7}}$


(iv) $y=\left(x^{2}+\frac{1}{x^{2}}\right)^{3}$
Sol :


(v) $y=\left(x-\frac{1}{x}\right)\left(x^{2}+\frac{1}{2 x}\right)$
Sol :
$y=\left(x-\frac{1}{2}\right)\left(x^{2}+\frac{1}{2 x}\right)$

$y=x^{3}+\frac{1}{2}-x-\frac{1}{2 x^{2}}$

$y=x^{3}+\frac{1}{2}-x-\frac{1}{2} x^{-2}$

Differentiating w.r.t x

$\frac{d y}{d x}=3 x^{2}+0-1-\frac{1}{2} \cdot(-2) \cdot x^{-3}$

$=3 x^{2}-1+\frac{1}{x^{3}}$


(vi) $y=\left(x+\frac{1}{x}\right)^{3}$
Sol :
$y=x^{3}+3 \cdot x^{2} \cdot \frac{1}{x}+3 \cdot x \cdot\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{x}\right)^{3}$

Differentiating w.r.t x

$\frac{d y}{d x}=3 x^{2}+3+3\left(-\frac{1}{x^{2}}\right)+(-3) x^{-4}$

$=3 x^{2}+3-\frac{3}{x^{2}}-\frac{3}{x^{4}}$


(vii) $y=\frac{a+b \cos x}{\sin x}$
Sol :
$y=\frac{a}{\sin x}+b \frac{\cos x}{\sin x}$

y=acosecx+bcotx

Differentiating w.r.t x

$\frac{d y}{d n}=-a \operatorname{cosec} x \operatorname{cot} x-b \cos x c^{2} x$


(viii) $y=\frac{b+c \tan x}{\sec x}$
Sol :
$y=b \cos x+c \frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}}$

y=bcosx+csinx

Differentiating w.r.t x

$\frac{d y}{dx}=-b \sin x+c \cos x$


(ix) x5(3-6x-9)
Sol :
y=3x5-6x-4

Differentiating w.r.t x

$\frac{d y}{dx}=3 \times 5 x^{4}-6(-4) \cdot x^{-5}$

$=15 x^{4}+\frac{24}{x^{5}}$


(x) x-3(5+3x)
Sol :


(xi) x-4(3-4x-5)
Sol :
y=x-4(3-4x-5)

y=3x-4-4x-9

Differentiating w.r.t x

$\frac{d y}{dx}=3 \cdot(-4) x^{-5}-4(-9) x^{-10}$

$=\frac{-12}{x^{5}}+\frac{36}{x^{10}}$


(xii) (5x3+3x-1)(x-1)
Sol :

Question 8

वक्र f(x)=2x6+x4-1 का x=1 पर स्पर्श रेखा की ढाल निकाले ।
[Find the slope of the tangent to the curve f(x)=2x6+x4-1 at x=1]
Sol :
f(x)=2x6+x4-1 

Differentiating w.r.t x

f '(x)=2×6x5+4x3-0 

'(x)=12x5+4x3 

स्पर्श रेखा की ढाल (x=1)='(1)
=12(1)5+4(1)3
=12+4
=16


Question 9

यदि f(x)=x2-9x+20 तो f '(x) निकाले और इससे f '(100) तथा $f^{\prime}\left(\frac{9}{2}\right)$ निकाले
Sol :
 f(x)=x2-9x+20

Differentiating w.r.t x

f '(x)=2x-9

f '(100)=2(100)-9
=200-9
=191

$f^{\prime}\left(\frac{9}{2}\right)=2\left(\frac{9}{2}\right)-9=0$

Question 10

यदि (If) $y=\frac{2-3 \cos x}{\sin x}, x=\frac{\pi}{4}$ पर $\frac{d y}{d x}$ निकाले (find $\frac{d y}{d x}$ at $\left.x=\frac{\pi}{4}\right)$
Sol :
$y=\frac{2-3 \cos x}{\sin x}$

$y=\frac{2}{\sin x}-\frac{3 \cos x}{\operatorname{sin} x}$

y=2cosec x-3cot x

Differentiating w.r.t x

$\frac{dy}{dx}=-2 \operatorname{cosec} x \cot x+3 \operatorname{cosec}^{2} x$

$x=\frac{\pi}{4}$ पर,

$\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{4}}=-2 \operatorname{cosec} \frac{\pi}{4} \cot \frac{\pi}{4}+3 \operatorname{cosec}^{2} \frac{\pi}{4}$

$=-2 \times \sqrt{2} \times 1+3(\sqrt{2})^{2}$

$=-2 \sqrt{2}+6=6-2 \sqrt{2}$


Question 11

यदि y=(x-a)(x-b) तो x का मान निकाले जिसके लिए $\frac{d y}{d x}=0$
Sol :
y=x2-bx-ax+ab

Differentiating w.r.t x

$\frac{dy}{dx}=2 x-b-a+0$

$\frac{dy}{dx}=2 x-(a+b)$

$\because \frac{d y}{d x}=0$

2x-(a+b)=0
2x=a+b

$x=\frac{a+b}{2}$


Question 12

यदि (If) $f(x)=\frac{x-4}{2 \sqrt{x}}$ , तो f '(4) निकााले (then '(4))
Sol :
$f(x)=\frac{x-4}{2 x^{1 / 2}}$

$=\frac{x}{2 x^{1 / 2}}-\frac{4}{2 x^{1 / 2}}$

$f(x)=\frac{1}{2} \sqrt{x}-2 x^{-1/ 2}$

Differentiating w.r.t x

$f^{\prime}(x)=\frac{1}{2} \times \frac{1}{2 \sqrt{x}}-2 \cdot\left(\frac{-1}{2}\right) x^{-3 / 2}$

$f^{\prime}(x)=\frac{1}{4 \sqrt{x}}+\frac{1}{x^{\frac{3}{2}}}$

$f^{\prime}(x)=\frac{1}{4 \sqrt{x}}+\frac{1}{x \sqrt{2}}$

x=4 , 

$f^{\prime}(4)=\frac{1}{4 \sqrt{4}}+\frac{1}{4 \sqrt{4}}$

$=\frac{1}{8}+\frac{1}{8}$

$=\frac{2}{8}=\frac{1}{4}$


Question 13

यदि (If) f(x)=1+x+x2+x3+...x50 , तो '(1) निकाले (then find '(1) )
Sol :
f(x)=1+x+x2+x3+...x50

Differentiating w.r.t x

f '(x)=1+2x+3x2+...50x49

x=1 पर,

'(x)=1+2(1)+3(1)2+...50(1)49
=1+2+3+...50
$=\frac{50 (50+1)}{2}$
=25×51
=1275

Question 14

यदि(If) $f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots+\frac{x^{2}}{2}+x+1$, साबित केर कि(prove that) '(1)=100'(0)
Sol :
$f(x)=\frac{x^{100}}{(100)}+\frac{x^{99}}{99}+\dots+\frac{x^{2}}{2}+x+1$

Differentiating w.r.t x

$f^{\prime}(x)=\frac{100 x^{99}}{100}+\frac{99 x^{98}}{99}+\dots \frac{2 x}{2}+1+0$

f '(x)=x99+x98+...+x+1

f '(x)=(1)99+(1)98+....+1+1=100

100f '(0)=100[(0)99+(0)98+....0+1]

=100×1
=100

f '(1)=100f '(0)

Question 15

यदि (If) y=cosecx+cotx, साबित करे कि(prove that) $\frac{d y}{d x}+y \operatorname{cosec} x=0$
Sol :
y=cosecx+cotx

Differentiating w.r.t x

$\frac{d y}{d x}=-\operatorname{cosec} x \cot x-\operatorname{cosec}^{2} x$

$\frac{d y}{dx}=-\operatorname{cosec} x(\cot x+\operatorname{cosec} x)$

$\frac{dy}{dx}=-y \operatorname{cosec} x$

$\frac{d y}{d x}+y \operatorname{cosec} x=0$


Question 16

यदि (If) $y=x+\frac{1}{x}$, साबित करे कि (prove that) $x^{2} \frac{d y}{d x}-x y+2=0$
Sol :
$y=x+\frac{1}{x}$

Differentiating w.r.t x

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

L.H.S
$x^{2} \frac{dy}{d x}-x y+2$

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

=x2-1-x2-1+2
=-2+2
=0

Question 17

यदि(If) $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ , साबित करे कि (show that) $2 x \frac{d y}{d x}+y=2 \sqrt{x}$
Sol :
$y=\sqrt{x}+\frac{1}{\sqrt{x}}$

$y=\sqrt{x}+\frac{1}{x^{1 / 2}}$

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

Differentiating w.r.t x

$\frac{d y}{d x}=\frac{1}{2 \sqrt{x}}+\left(-\frac{1}{2}\right) x^{-3 / 2}$

$\frac{d y}{dx}=\frac{1}{2 \sqrt{x}}-\frac{1}{2 x^{3} / 2}$

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


L.H.S
$2 x \frac{d y}{dx}+y$

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

$=\sqrt{x}-\frac{1}{\sqrt{x}}+\sqrt{x}+\frac{1}{\sqrt{x}}$

$=2 \sqrt{x}$=R.H.S

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