Advertisements
Advertisements
प्रश्न
\[\frac{1 + \log x}{1 - \log x}\]
Advertisements
उत्तर
\[\text{ Let } u = 1 + \log x; v = 1 - \log x\]
\[\text{ Then }, u' = \frac{1}{x}; v' = \frac{- 1}{x}\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{1 + \log x}{1 - \log x} \right) = \frac{\left( 1 - \log x \right)\left( \frac{1}{x} \right) - \left( 1 + \log x \right)\left( \frac{- 1}{x} \right)}{\left( 1 - \log x \right)^2}\]
\[ = \frac{1 - \log x + 1 + \log x}{x \left( 1 - \log x \right)^2}\]
\[ = \frac{2}{x \left( 1 - \log x \right)^2}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x–3 (5 + 3x).
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):
(x + a)
Find the derivative of f (x) x at x = 1
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 2}{3x + 5}\]
\[\sqrt{2 x^2 + 1}\]
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan2 x
\[\tan \sqrt{x}\]
ex log a + ea long x + ea log a
(2x2 + 1) (3x + 2)
log3 x + 3 loge x + 2 tan x
2 sec x + 3 cot x − 4 tan x
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x3 sin x
(x3 + x2 + 1) sin x
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log 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
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
(ax2 + cot x)(p + q cos x)
