Advertisements
Advertisements
प्रश्न
Differentiate of the following from first principle:
sin (x + 1)
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f\left( x \right) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( \sin \left( x + 1 \right) \right) = \lim_{h \to 0} \frac{\sin \left( x + h + 1 \right) - \sin \left( x + 1 \right)}{h}\]
\[\text{ We know }:\]
\[\sin C - \sin D = 2 \cos \left( \frac{C + D}{2} \right) \sin \left( \frac{C - D}{2} \right)\]
\[ = \lim_{h \to 0} \frac{2 \cos \left( \frac{x + h + 1 + x + 1}{2} \right) \sin \left( \frac{x + h + 1 - x - 1}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{2 \cos \left( \frac{2x + h + 2}{2} \right) \sin \left( \frac{h}{2} \right)}{h}\]
\[ = 2 \lim_{h \to 0} \cos \left( \frac{2x + h + 2}{2} \right) \lim_{h \to 0} \frac{\sin \left( \frac{h}{2} \right)}{\frac{h}{2}} \times \frac{1}{2}\]
\[ = 2 \cos \left( x + 1 \right) \times \frac{1}{2}\]
\[ = \cos \left( x + 1 \right)\]
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):
`(ax + b)/(cx + d)`
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):
`cos x/(1 + sin x)`
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):
`(sec x - 1)/(sec x + 1)`
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):
sinn x
Find the derivative of f (x) = 99x at x = 100
Find the derivative of f (x) = cos x at x = 0
Find the derivative of f (x) = tan x at x = 0
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 + 1}{x}\]
\[\frac{x + 1}{x + 2}\]
\[\sqrt{2 x^2 + 1}\]
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
x2 ex
x4 − 2 sin x + 3 cos x
log3 x + 3 loge x + 2 tan x
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
cos (x + a)
x2 ex log x
xn loga x
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
x5 ex + x6 log x
(1 − 2 tan x) (5 + 4 sin x)
x3 ex cos x
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x}{\sin^n x}\]
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 \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Find the derivative of x2 cosx.
(ax2 + cot x)(p + q cos x)
