Advertisements
Advertisements
प्रश्न
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
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 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):
`(px+ q) (r/s + s)`
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/(ax^2 + bx + c)`
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):
`(px^2 +qx + r)/(ax +b)`
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 at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 + 1}{x}\]
k xn
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
sin (x + 1)
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
x2 ex
tan2 x
log3 x + 3 loge x + 2 tan x
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
(x sin x + cos x ) (ex + x2 log x)
x4 (5 sin x − 3 cos x)
\[\frac{e^x}{1 + x^2}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
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\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
