Advertisements
Advertisements
प्रश्न
Find the derivative of f (x) = cos x at x = 0
Advertisements
उत्तर
We have:
\[f'(x) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}\]
\[ = \lim_{h \to 0} \frac{f(h) - f(0)}{h}\]
\[ = \lim_{h \to 0} \frac{\cosh - \cos0}{h}\]
\[ = \lim_{h \to 0} \frac{\cosh - 1}{h}\]
\[ {= \lim}_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{h}\]
\[ {= \lim_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{\frac{h^2}{4}}}_{} \times \frac{h}{4}\]
\[ = {= \lim_{h \to 0} - 1}_{} \times \frac{h}{2}\]
\[ = 0\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x–3 (5 + 3x).
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):
`(1 + 1/x)/(1- 1/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):
`(px^2 +qx + r)/(ax +b)`
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):
`4sqrtx - 2`
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 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):
`(sin x + cos x)/(sin x - cos x)`
Find the derivative of f (x) = tan x at x = 0
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 1}{x + 2}\]
Differentiate each of the following from first principle:
e−x
Differentiate each of the following from first principle:
sin x + cos x
\[\tan \sqrt{x}\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
(1 +x2) cos x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
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.
(3x2 + 2)2
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.
(x + 2) (x + 3)
(ax + b) (a + d)2
(ax + b)n (cx + d)n
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{x}{\sin^n x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] 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
Find the derivative of 2x4 + x.
Find the derivative of x2 cosx.
