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
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
Find the derivative of x5 (3 – 6x–9).
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+ q) (r/s + s)`
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):
`(sin x + cos x)/(sin x - cos 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 + a))/ cos x`
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) = tan x at x = 0
Find the derivative of the following function at the indicated point:
\[\frac{x^2 + 1}{x}\]
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
x2 ex log x
sin x cos x
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
(1 +x2) cos x
sin2 x
\[e^x \log \sqrt{x} \tan x\]
x−4 (3 − 4x−5)
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
(ax + b)n (cx + d)n
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{x + \cos x}{\tan x}\]
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 derivative of f (x) = 3 |2 + x| at x = −3.
Find the derivative of x2 cosx.
