Advertisements
Advertisements
Question
x ex
Advertisements
Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( x e^x \right) = \lim_{h \to 0} \frac{(x + h ) e^{(x + h)} - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{(x + h) e^x e^h - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x e^h + h e^x e^h - x e^x}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x e^h - x e^x}{h} + \lim_{h \to 0} \frac{h e^x e^h}{h}\]
\[ = \lim_{h \to 0} \frac{x e^x \left( e^h - 1 \right)}{h} + \lim_{h \to 0} e^x e^h \]
\[ = x e^x \left( 1 \right) + e^x \left( e^0 \right)\]
\[ = x e^x + e^x\]
APPEARS IN
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):
`4sqrtx - 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):
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):
sinn x
Find the derivative of f (x) = 99x at x = 100
\[\frac{1}{x^3}\]
k xn
(x + 2)3
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
\[\tan \sqrt{x}\]
ex log a + ea long x + ea log a
(2x2 + 1) (3x + 2)
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
(x sin x + cos x) (x cos x − sin x)
(x sin x + cos x ) (ex + x2 log x)
(1 − 2 tan x) (5 + 4 sin x)
\[e^x \log \sqrt{x} \tan x\]
x4 (5 sin x − 3 cos x)
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(x + 2) (x + 3)
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{1}{a x^2 + bx + c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
(ax2 + cot x)(p + q cos x)
