Advertisements
Advertisements
प्रश्न
\[\sqrt{2 x^2 + 1}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sqrt{2 \left( x + h \right)^2 + 1} - \sqrt{2 x^2 + 1}}{h}\]
\[ = \lim_{h \to 0} \frac{\sqrt{2 x^2 + 2 h^2 + 4xh + 1} - \sqrt{2 x^2 + 1}}{h}\]
\[ = \lim_{h \to 0} \frac{\sqrt{2 x^2 + 2 h^2 + 4xh + 1} - \sqrt{2 x^2 + 1}}{h} \times \frac{\sqrt{2 x^2 + 2 h^2 + 4xh + 1} + \sqrt{2 x^2 + 1}}{\sqrt{2 x^2 + 2 h^2 + 4xh + 1} + \sqrt{2 x^2 + 1}}\]
\[ = \lim_{h \to 0} \frac{2 x^2 + 2 h^2 + 4xh + 1 - 2 x^2 - 1}{h\left( \sqrt{2 x^2 + 2 h^2 + 4xh + 1} + \sqrt{2 x^2 + 1} \right)}\]
\[ = \lim_{h \to 0} \frac{h\left( 2h + 4x \right)}{h\left( \sqrt{2 x^2 + 2 h^2 + 4xh + 1} + \sqrt{2 x^2 + 1} \right)}\]
\[ = \lim_{h \to 0} \frac{\left( 2h + 4x \right)}{\left( \sqrt{2 x^2 + 2 h^2 + 4xh + 1} + \sqrt{2 x^2 + 1} \right)}\]
\[ = \frac{4x}{\sqrt{2 x^2 + 1} + \sqrt{2 x^2 + 1}}\]
\[ = \frac{4x}{2\sqrt{2 x^2 + 1}}\]
\[ = \frac{2x}{\sqrt{2 x^2 + 1}}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x at x = 1.
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):
`(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 f (x) = 99x at x = 100
Find the derivative of f (x) = cos x at x = 0
\[\frac{x + 2}{3x + 5}\]
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
e3x
x ex
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
x2 sin x
tan2 x
\[\cos \sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\frac{2 x^2 + 3x + 4}{x}\]
2 sec x + 3 cot x − 4 tan x
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x2 ex log x
sin x cos x
(1 +x2) cos x
x3 ex cos x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
(2x2 − 3) sin x
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
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.
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Find the derivative of x2 cosx.
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
