Advertisements
Advertisements
प्रश्न
x ex
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( x e^x \right) = \lim_{h \to 0} \frac{(x + h ) e^{(x + h)} - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{(x + h) e^x e^h - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x e^h + h e^x e^h - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x e^h - x e^x}{h} + \lim_{h \to 0} \frac{h e^x e^h}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x \left( e^h - 1 \right)}{h} + \lim_{h \to 0} e^x e^h \]
\[ = x e^x \left( 1 \right) + e^x \left( e^0 \right)\]
\[ = x e^x + e^x\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
Find the derivative of x–3 (5 + 3x).
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):
`(px+ q) (r/s + s)`
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):
`(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):
`(sec x - 1)/(sec 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):
sinn 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:
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
k xn
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
− x
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
tan2 x
\[\cos \sqrt{x}\]
ex log a + ea long x + ea log a
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x3 sin x
xn loga x
sin x cos x
(2x2 − 3) sin x
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{x}{\sin^n x}\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
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\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Find the derivative of x2 cosx.
Find the derivative of f(x) = tan(ax + b), by first principle.
