Advertisements
Advertisements
Question
Differentiate each of the following from first principle:
x2 ex
Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( x^2 e^x \right) = \lim_{h \to 0} \frac{(x + h )^2 e^{(x + h)} - x^2 e^x}{h}\]
\[ = \lim_{h \to 0} \frac{( x^2 + 2xh + h^2 ) e^x e^h - x^2 e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 e^x e^h + 2xh e^x e^h + h^2 e^x e^h - x^2 e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 e^x e^h - x^2 e^x}{h} + \lim_{h \to 0} \frac{2 x h e^x e^h}{h} + \lim_{h \to 0} \frac{h^2 e^x e^h}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 e^x \left( e^h - 1 \right)}{h} + \lim_{h \to 0} 2 x e^x e^h + \lim_{h \to 0} h e^x e^h \]
\[ = x^2 e^x \left( 1 \right) + 2x e^x \left( 1 \right) + 0\]
\[ = x^2 e^x + 2x e^x \]
\[ = \left( x^2 + 2x \right) e^x \]
APPEARS IN
RELATED QUESTIONS
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):
(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 f (x) = 99x at x = 100
Find the derivative of f (x) = tan x at x = 0
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:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
2 sec x + 3 cot x − 4 tan x
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
sin x cos x
x3 ex cos x
x−3 (5 + 3x)
(ax + b)n (cx + d)n
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{x}{\sin^n x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
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:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
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 \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
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
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
Find the derivative of 2x4 + x.
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
