Advertisements
Advertisements
Question
\[\frac{x + e^x}{1 + \log x}\]
Advertisements
Solution
\[\text{ Let } u = x + e^x ; v = 1 + \log x\]
\[\text{ Then }, u' = 1 + e^x ; v' = \frac{1}{x}\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{x + e^x}{1 + \log x} \right) = \frac{\left( 1 + \log x \right)\left( 1 + e^x \right) - \left( x + e^x \right)\left( \frac{1}{x} \right)}{(1 + \log x )^2}\]
\[ = \frac{x + x e^x + x \log x + x \log x e^x - x - e^x}{x(1 + \log x )^2}\]
\[ = \frac{x \log x + x \log x e^x - e^x + x e^x}{x(1 + \log x )^2}\]
\[ = \frac{x \log x \left( 1 + e^x \right) - e^x \left( 1 - x \right)}{x(1 + \log x )^2}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
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 + cos x)/(sin x - 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):
`(a + bsin x)/(c + dcosx)`
\[\frac{2}{x}\]
k xn
Differentiate of the following from first principle:
e3x
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:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
3x + x3 + 33
x2 ex log x
xn loga x
\[\frac{2^x \cot x}{\sqrt{x}}\]
(1 +x2) cos x
x−3 (5 + 3x)
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)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
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\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
