Advertisements
Advertisements
प्रश्न
Differentiate each of the following from first principle:
x2 ex
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( x^2 e^x \right) = \lim_{h \to 0} \frac{(x + h )^2 e^{(x + h)} - x^2 e^x}{h}\]
\[ = \lim_{h \to 0} \frac{( x^2 + 2xh + h^2 ) e^x e^h - x^2 e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 e^x e^h + 2xh e^x e^h + h^2 e^x e^h - x^2 e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 e^x e^h - x^2 e^x}{h} + \lim_{h \to 0} \frac{2 x h e^x e^h}{h} + \lim_{h \to 0} \frac{h^2 e^x e^h}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 e^x \left( e^h - 1 \right)}{h} + \lim_{h \to 0} 2 x e^x e^h + \lim_{h \to 0} h e^x e^h \]
\[ = x^2 e^x \left( 1 \right) + 2x e^x \left( 1 \right) + 0\]
\[ = x^2 e^x + 2x e^x \]
\[ = \left( x^2 + 2x \right) e^x \]
APPEARS IN
संबंधित प्रश्न
Find the derivative of `2x - 3/4`
Find the derivative of (5x3 + 3x – 1) (x – 1).
Find the derivative of x5 (3 – 6x–9).
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):
(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):
`(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):
sinn x
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) x at x = 1
Find the derivative of the following function at the indicated point:
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 + 1}{x}\]
\[\sqrt{2 x^2 + 1}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
tan (2x + 1)
tan 2x
\[\tan \sqrt{x}\]
\[\frac{2 x^2 + 3x + 4}{x}\]
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.
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
(1 − 2 tan x) (5 + 4 sin x)
x−3 (5 + 3x)
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
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 \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
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
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of x2 cosx.
