Advertisements
Advertisements
प्रश्न
\[\frac{x + e^x}{1 + \log x}\]
Advertisements
उत्तर
\[\text{ Let } u = x + e^x ; v = 1 + \log x\]
\[\text{ Then }, u' = 1 + e^x ; v' = \frac{1}{x}\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{x + e^x}{1 + \log x} \right) = \frac{\left( 1 + \log x \right)\left( 1 + e^x \right) - \left( x + e^x \right)\left( \frac{1}{x} \right)}{(1 + \log x )^2}\]
\[ = \frac{x + x e^x + x \log x + x \log x e^x - x - e^x}{x(1 + \log x )^2}\]
\[ = \frac{x \log x + x \log x e^x - e^x + x e^x}{x(1 + \log x )^2}\]
\[ = \frac{x \log x \left( 1 + e^x \right) - e^x \left( 1 - x \right)}{x(1 + \log x )^2}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x at x = 1.
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 `2x - 3/4`
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):
`(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):
cosec x cot 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):
`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):
sinn 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) = tan x at x = 0
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{1}{\sqrt{x}}\]
\[\frac{x^2 + 1}{x}\]
\[\frac{x + 1}{x + 2}\]
\[\sqrt{2 x^2 + 1}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
tan (2x + 1)
\[\tan \sqrt{x}\]
\[\tan \sqrt{x}\]
\[\frac{2 x^2 + 3x + 4}{x}\]
cos (x + a)
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
x3 sin x
xn loga x
x5 ex + x6 log x
(1 − 2 tan x) (5 + 4 sin 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
(ax + b)n (cx + d)n
\[\frac{x}{1 + \tan x}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
