Advertisements
Advertisements
प्रश्न
\[\frac{2^x \cot x}{\sqrt{x}}\]
Advertisements
उत्तर
\[\frac{2^x \cot x}{\sqrt{x}} = 2^x \cot x \left( x^\frac{- 1}{2} \right)\]
\[\text{ Let } u = 2^x ; v = \cot x; w = x^\frac{- 1}{2} \]
\[\text{ Then }, u' = 2^x \log 2; v' = - {cosec}^2 x; w' = \frac{- 1}{2} x^\frac{- 3}{2} \]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uvw \right) = u'vw + uv'w + uvw'\]
\[\frac{d}{dx}\left[ 2^x \cot x \left( x^\frac{- 1}{2} \right) \right] = 2^x \log 2 . \cot x . x^\frac{- 1}{2} + 2^x \left( - {cosec}^2 x \right) x^\frac{- 1}{2} + 2^x \cot x\left( \frac{- 1}{2} x^\frac{- 3}{2} \right)\]
\[ = 2^x \log 2 . \cot x . \frac{1}{\sqrt{x}} + 2^x \left( - {cosec}^2 x \right)\frac{1}{\sqrt{x}} + 2^x \cot x\left( \frac{- 1}{2x\sqrt{x}} \right)\]
\[ = \frac{2^x}{\sqrt{x}}\left( \log 2 . \cot x - {cosec}^2 x - \frac{\cot x}{2x} \right)\]
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 `2/(x + 1) - x^2/(3x -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):
`(ax + b)/(px^2 + qx + r)`
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)
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = 99x at x = 100
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 =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{1}{\sqrt{x}}\]
x2 + x + 3
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
tan 2x
\[\sin \sqrt{2x}\]
\[\tan \sqrt{x}\]
3x + x3 + 33
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
x2 sin x log x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{x + \cos x}{\tan x}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of f(x) = tan(ax + b), by first principle.
