Advertisements
Advertisements
Question
\[\frac{1 + \log x}{1 - \log x}\]
Advertisements
Solution
\[\text{ Let } u = 1 + \log x; v = 1 - \log x\]
\[\text{ Then }, u' = \frac{1}{x}; v' = \frac{- 1}{x}\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{1 + \log x}{1 - \log x} \right) = \frac{\left( 1 - \log x \right)\left( \frac{1}{x} \right) - \left( 1 + \log x \right)\left( \frac{- 1}{x} \right)}{\left( 1 - \log x \right)^2}\]
\[ = \frac{1 - \log x + 1 + \log x}{x \left( 1 - \log x \right)^2}\]
\[ = \frac{2}{x \left( 1 - \log x \right)^2}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
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 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 + bsin x)/(c + dcosx)`
Find the derivative of f (x) = 99x at x = 100
\[\frac{1}{\sqrt{x}}\]
\[\frac{1}{x^3}\]
\[\frac{x + 2}{3x + 5}\]
\[\frac{1}{\sqrt{3 - x}}\]
Differentiate each of the following from first principle:
e−x
x ex
Differentiate of the following from first principle:
sin (x + 1)
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
3x + x3 + 33
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
xn loga x
\[e^x \log \sqrt{x} \tan x\]
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
