Advertisements
Advertisements
प्रश्न
\[\frac{1}{\sqrt{3 - x}}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{1}{\sqrt{3 - x - h}} - \frac{1}{\sqrt{3 - x}}}{h}\]
\[ = \lim_{h \to 0} \frac{\left( \sqrt{3 - x} - \sqrt{3 - x - h} \right)}{h\sqrt{3 - x}\sqrt{3 - x - h}}\]
\[ = \lim_{h \to 0} \frac{\left( \sqrt{3 - x} - \sqrt{3 - x - h} \right)}{h\sqrt{3 - x}\sqrt{3 - x - h}} \times \frac{\left( \sqrt{3 - x} + \sqrt{3 - x - h} \right)}{\left( \sqrt{3 - x} + \sqrt{3 - x - h} \right)}\]
\[ = \lim_{h \to 0} \frac{\left( 3 - x - 3 + x + h \right)}{h\sqrt{3 - x}\sqrt{3 - x - h}\left( \sqrt{3 - x} + \sqrt{3 - x - h} \right)}\]
\[ = \lim_{h \to 0} \frac{h}{h\sqrt{3 - x}\sqrt{3 - x - h}\left( \sqrt{3 - x} + \sqrt{3 - x - h} \right)}\]
\[ = \lim_{h \to 0} \frac{1}{\sqrt{3 - x}\sqrt{3 - x - h}\left( \sqrt{3 - x} + \sqrt{3 - x - h} \right)}\]
\[ = \frac{1}{\sqrt{3 - x}\sqrt{3 - x - 0}\left( \sqrt{3 - x} + \sqrt{3 - x - 0} \right)}\]
\[ = \frac{1}{\left( 3 - x \right) \left( 2\sqrt{3 - x} \right)}\]
\[ = \frac{1}{2 \left( 3 - x \right)^\frac{3}{2}}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
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)`
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 f (x) = 3x at x = 2
Find the derivative of f (x) x at x = 1
\[\frac{2x + 3}{x - 2}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
e3x
\[\sin \sqrt{2x}\]
\[\cos \sqrt{x}\]
\[\frac{2 x^2 + 3x + 4}{x}\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
cos (x + a)
(x3 + x2 + 1) sin x
logx2 x
(2x2 − 3) sin x
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.
(3x2 + 2)2
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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
(ax + b)n (cx + d)n
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{1}{a x^2 + bx + c}\]
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 \[\frac{\pi}{2}\] 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 (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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.
