Advertisements
Advertisements
प्रश्न
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
Advertisements
उत्तर
\[\text{ Let } u = \sqrt{a} + \sqrt{x}; v = \sqrt{a} - \sqrt{x}\]
\[\text{ Then }, u' = \frac{1}{2\sqrt{x}}; v' = \frac{- 1}{2\sqrt{x}}\]
\[\text{ Using thequotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}} \right) = \frac{\left( \sqrt{a} - \sqrt{x} \right)\frac{1}{2\sqrt{x}} - \left( \sqrt{a} + \sqrt{x} \right)\left( \frac{- 1}{2\sqrt{x}} \right)}{\left( \sqrt{a} - \sqrt{x} \right)^2}\]
\[ = \frac{\sqrt{a} - \sqrt{x} + \sqrt{a} + \sqrt{x}}{2\sqrt{x} \left( \sqrt{a} - \sqrt{x} \right)^2}\]
\[ = \frac{2\sqrt{a}}{2\sqrt{x} \left( \sqrt{a} - \sqrt{x} \right)^2}\]
\[ = \frac{\sqrt{a}}{\sqrt{x} \left( \sqrt{a} - \sqrt{x} \right)^2}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x at x = 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):
(ax + b) (cx + d)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):
`(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):
`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):
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):
`(a + bsin x)/(c + dcosx)`
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 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) = cos x at x = 0
Find the derivative of f (x) = tan x at x = 0
\[\frac{x^2 + 1}{x}\]
\[\frac{x + 1}{x + 2}\]
x2 + x + 3
(x2 + 1) (x − 5)
Differentiate each of the following from first principle:
e−x
x ex
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
tan2 x
\[\tan \sqrt{x}\]
3x + x3 + 33
ex log a + ea long x + ea log a
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x2 sin x log x
(x sin x + cos x) (x cos x − sin x)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{3^x}{x + \tan x}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
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 \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
(ax2 + cot x)(p + q cos x)
`(a + b sin x)/(c + d cos x)`
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
