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
Find the derivative of 99x at x = 100.
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)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):
`(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):
`a/x^4 = b/x^2 + 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):
`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):
sin (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):
`cos x/(1 + sin x)`
Find the derivative of f (x) = 3x at x = 2
\[\frac{2}{x}\]
\[\frac{x^2 - 1}{x}\]
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
x ex
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
x3 sin x
x5 ex + x6 log x
(1 − 2 tan x) (5 + 4 sin x)
(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.
(x + 2) (x + 3)
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)
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{1}{a x^2 + bx + c}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Find the derivative of x2 cosx.
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
