Advertisements
Advertisements
प्रश्न
\[\sqrt{2 x^2 + 1}\]
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 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):
`(ax + b)/(px^2 + qx + r)`
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):
`(sin x + cos x)/(sin x - 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):
sinn 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 the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{x + 1}{x + 2}\]
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
eax + b
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
tan (2x + 1)
log3 x + 3 loge x + 2 tan x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{2 x^2 + 3x + 4}{x}\]
cos (x + a)
(1 − 2 tan x) (5 + 4 sin x)
(1 +x2) cos x
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Find the derivative of 2x4 + x.
