Advertisements
Advertisements
प्रश्न
\[\frac{e^x + \sin x}{1 + \log x}\]
Advertisements
उत्तर
\[\text{ Let } u = e^x + \sin x; v = 1 + \log x\]
\[\text{ Then }, u' = e^x + \cos x; v' = \frac{1}{x}\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{e^x + \sin x}{1 + \log x} \right) = \frac{\left( 1 + \log x \right)\left( e^x + \cos x \right) - \left( e^x + \sin x \right)\left( \frac{1}{x} \right)}{\left( 1 + \log x \right)^2}\]
\[ = \frac{x\left( 1 + \log x \right)\left( e^x + \cos x \right) - \left( e^x + \sin x \right)}{x \left( 1 + \log x \right)^2}\]
APPEARS IN
संबंधित प्रश्न
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):
`4sqrtx - 2`
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)n
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)n (cx + d)m
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):
x4 (5 sin x – 3 cos x)
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) x at x = 1
Find the derivative of f (x) = cos x at x = 0
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 =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 + 1}{x}\]
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
\[3^{x^2}\]
tan2 x
tan (2x + 1)
\[\sin \sqrt{2x}\]
3x + x3 + 33
(2x2 + 1) (3x + 2)
2 sec x + 3 cot x − 4 tan x
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
x3 ex
x5 ex + x6 log x
sin2 x
(2x2 − 3) sin x
x−4 (3 − 4x−5)
(ax + b)n (cx + d)n
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
Mark the correct alternative in of the following:
If\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Find the derivative of f(x) = tan(ax + b), by first principle.
