Advertisements
Advertisements
Question
\[\frac{2^x \cot x}{\sqrt{x}}\]
Advertisements
Solution
\[\frac{2^x \cot x}{\sqrt{x}} = 2^x \cot x \left( x^\frac{- 1}{2} \right)\]
\[\text{ Let } u = 2^x ; v = \cot x; w = x^\frac{- 1}{2} \]
\[\text{ Then }, u' = 2^x \log 2; v' = - {cosec}^2 x; w' = \frac{- 1}{2} x^\frac{- 3}{2} \]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uvw \right) = u'vw + uv'w + uvw'\]
\[\frac{d}{dx}\left[ 2^x \cot x \left( x^\frac{- 1}{2} \right) \right] = 2^x \log 2 . \cot x . x^\frac{- 1}{2} + 2^x \left( - {cosec}^2 x \right) x^\frac{- 1}{2} + 2^x \cot x\left( \frac{- 1}{2} x^\frac{- 3}{2} \right)\]
\[ = 2^x \log 2 . \cot x . \frac{1}{\sqrt{x}} + 2^x \left( - {cosec}^2 x \right)\frac{1}{\sqrt{x}} + 2^x \cot x\left( \frac{- 1}{2x\sqrt{x}} \right)\]
\[ = \frac{2^x}{\sqrt{x}}\left( \log 2 . \cot x - {cosec}^2 x - \frac{\cot x}{2x} \right)\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
Find the derivative of x at x = 1.
Find the derivative of `2x - 3/4`
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):
`(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):
`1/(ax^2 + bx + c)`
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) x at x = 1
Find the derivative of the following function at the indicated point:
\[\frac{x^2 + 1}{x}\]
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
sin (x + 1)
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x3 sin x
x3 ex
xn tan x
x5 ex + x6 log x
(x sin x + cos x) (x cos x − sin x)
(1 − 2 tan x) (5 + 4 sin x)
\[e^x \log \sqrt{x} \tan x\]
x−4 (3 − 4x−5)
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)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x}{\sin^n x}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
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
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
