Advertisements
Advertisements
प्रश्न
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
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 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):
(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):
`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):
(ax + b)n
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 f (x) = 99x at x = 100
\[\frac{2}{x}\]
\[\frac{1}{\sqrt{x}}\]
\[\frac{1}{\sqrt{3 - x}}\]
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
x cos x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
x3 ex
x5 ex + x6 log x
(x sin x + cos x ) (ex + x2 log x)
(1 − 2 tan x) (5 + 4 sin x)
(1 +x2) cos x
(2x2 − 3) sin x
x−3 (5 + 3x)
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.
(3x2 + 2)2
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Find the derivative of 2x4 + x.
