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 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):
`(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):
(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):
`cos x/(1 + sin 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):
sinn x
Find the derivative of f (x) = 3x at x = 2
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{1}{x^3}\]
(x + 2)3
Differentiate each of the following from first principle:
e−x
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan2 x
\[\tan \sqrt{x}\]
(2x2 + 1) (3x + 2)
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
(1 − 2 tan x) (5 + 4 sin x)
\[e^x \log \sqrt{x} \tan x\]
(2x2 − 3) sin x
x5 (3 − 6x−9)
x−3 (5 + 3x)
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{x}{1 + \tan x}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
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\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to
`(a + b sin x)/(c + d cos x)`
