Advertisements
Advertisements
प्रश्न
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
उत्तर
\[\frac{d}{dx}\left( e^\sqrt{2x} \right) = \lim_{h \to 0} \frac{e^\sqrt{2(x + h)} - e^\sqrt{2x}}{h}\]
\[ = 2 \lim_{h \to 0} \frac{e^\sqrt{2x + 2h} - e^\sqrt{2x}}{2x + 2h - 2x}\]
\[ = 2 \lim_{h \to 0} \frac{e^\sqrt{2x} \left( e^\sqrt{2x + 2h} - \sqrt{2x} - 1 \right)}{\left( \sqrt{2x + 2h} \right)^2 - \left( \sqrt{2x} \right)^2}\]
\[ = 2 e^\sqrt{2x} \lim_{h \to 0} \frac{e^\sqrt{2x + 2h} - \sqrt{2x} - 1}{\left( \sqrt{2x + 2h} - \sqrt{2x} \right)\left( \sqrt{2x + 2h} + \sqrt{2x} \right)}\]
\[ = 2 e^\sqrt{2x} \lim_{h \to 0} \frac{e^\sqrt{2x + 2h} - \sqrt{2x} - 1}{\left( \sqrt{2x + 2h} - \sqrt{2x} \right)} \lim_{h \to 0} \frac{1}{\left( \sqrt{2x + 2h} + \sqrt{2x} \right)}\]
\[ = 2 e^\sqrt{2x} \left( 1 \right)\frac{1}{2\sqrt{2x}}\]
\[ = \frac{e^\sqrt{2x}}{\sqrt{2x}}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x5 (3 – 6x–9).
Find the derivative of x–4 (3 – 4x–5).
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):
`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):
(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 + a)
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 at the indicated point:
\[\frac{2}{x}\]
Differentiate each of the following from first principle:
e−x
x ex
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
x3 sin x
xn loga x
sin x cos x
(1 +x2) cos x
sin2 x
x4 (5 sin x − 3 cos x)
(2x2 − 3) sin x
x−4 (3 − 4x−5)
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
(ax + b) (a + d)2
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
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 \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
