Advertisements
Advertisements
प्रश्न
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Advertisements
उत्तर
\[\text{ Case } 1:\]
\[x > 0\]
\[|x| = x\]
\[\text{ Thus, we have }:\]
\[\frac{d}{dx}\left( x|x| \right) = \frac{d}{dx}\left( x . x \right) = \frac{d}{dx}\left( x^2 \right) = 2x \left( 1 \right)\]
\[\text{ Case } 2:\]
\[x < 0\]
\[|x| = - x\]
\[\text{ Thus, we have }:\]
\[\frac{d}{dx}\left( x|x| \right) = \frac{d}{dx}\left( x . \left( - x \right) \right) = \frac{d}{dx}\left( - x^2 \right) = - 2x \left( 2 \right)\]
\[\text{ From } (1) \text{ and } (2), \text{ we have }:\]
\[\frac{d}{dx}\left( x|x| \right) = \binom{2x, if x > 0}{ - 2x, if x < 0}\]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x at x = 1.
Find the derivative of (5x3 + 3x – 1) (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):
(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):
cosec x cot 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)
(x + 2)3
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
tan 2x
\[\sqrt{\tan x}\]
\[\tan \sqrt{x}\]
log3 x + 3 loge x + 2 tan x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
2 sec x + 3 cot x − 4 tan x
cos (x + a)
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
x5 ex + x6 log x
(x sin x + cos x) (x cos x − sin x)
logx2 x
x3 ex cos x
x−4 (3 − 4x−5)
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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{x}{\sin^n x}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
`(a + b sin x)/(c + d cos x)`
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
