Advertisements
Advertisements
प्रश्न
Differentiate of the following from first principle:
x sin x
Advertisements
उत्तर
\[ = \lim_{h \to 0} \frac{\left( x + h \right) \sin\left( x + h \right) - x \sin x}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h \right)\left( \sin x \cos h + \cos x \sin h \right) - x \sin x}{h}\]
\[ = \lim_{h \to 0} \frac{x \sin x \cos h + x \cos x \sin h + h \sin x \cos h + h \cos x \sin h}{h}\]
\[ = \lim_{h \to 0} \frac{x \sin x \cos h - x \sin x + x \cos x \sin h + h \sin x \cos h + h \cos x \sin h - x \sin x}{h}\]
\[ = x \sin x \lim_{h \to 0} \frac{\left( \cos h - 1 \right)}{h} + x \cos x \lim_{h \to 0} \frac{\sin h}{h} + \sin x \lim_{h \to 0} \cos h + \cos x \lim_{h \to 0} \sin h\]
\[ = x \sin x \lim_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{\frac{h^2}{4}} \times \frac{h}{4} + x \cos x \left( 1 \right) + \sin x \left( 1 \right) + \cos x \left( 0 \right)\]
\[ = x \sin x \times \frac{- h}{2} + x \cos x \left( 1 \right) + \sin x \left( 1 \right) + \cos x \left( 0 \right)\]
\[ = - 2x \sin x \left( \frac{1}{2} \right)\left( 0 \right) + x \cos x + \sin x \]
\[ = x \cos x + \sin x \]
\[ \]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
(x + a)
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
(ax + b)n (cx + d)m
Find the derivative of f (x) x at x = 1
\[\frac{x + 1}{x + 2}\]
\[\frac{x + 2}{3x + 5}\]
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
x ex
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
sin (x + 1)
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
tan 2x
\[\sqrt{\tan x}\]
ex log a + ea long x + ea log a
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
cos (x + a)
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
xn tan x
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
sin2 x
logx2 x
x−4 (3 − 4x−5)
x−3 (5 + 3x)
\[\frac{x}{1 + \tan x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
Mark the correct alternative in of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
`(a + b sin x)/(c + d cos x)`
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
