Advertisements
Advertisements
प्रश्न
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Advertisements
उत्तर
\[\text{ Case } 1:\]
\[x > 0\]
\[\left| x \right| = x\]
\[\left( x + \left| x \right| \right)\left| x \right|\]
\[ = \left( x + x \right)x\]
\[ = 2 x^2 \]
\[\frac{d}{dx}\left[ \left( x + \left| x \right| \right)\left| x \right| \right] = \frac{d}{dx}\left( 2 x^2 \right) = 4x \left( 1 \right)\]
\[\text{ Case } 2:\]
\[x < 0\]
\[\left| x \right| = - x\]
\[\left( x + \left| x \right| \right)\left| x \right|\]
\[ = \left( x - x \right)x\]
\[ = 0\]
\[\frac{d}{dx}\left[ \left( x + \left| x \right| \right)\left| x \right| \right] = \frac{d}{dx}\left( 0 \right) = 0 \left( 2 \right)\]
\[\text{ From } (1) \text{and} (2), \text{ we have}:\]
\[\frac{d}{dx}\left[ \left( x + \left| x \right| \right)\left| x \right| \right] = \binom{4x, if x > 0}{0, if x < 0}\]
\[\]
APPEARS IN
संबंधित प्रश्न
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 (cx + d)m
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):
`(a + bsin x)/(c + dcosx)`
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
\[\frac{x^2 - 1}{x}\]
k xn
Differentiate of the following from first principle:
e3x
x ex
tan (2x + 1)
tan 2x
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
(x sin x + cos x) (x cos x − sin x)
logx2 x
\[e^x \log \sqrt{x} \tan x\]
x3 ex cos x
(2x2 − 3) sin x
x−4 (3 − 4x−5)
(ax + b) (a + d)2
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
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 each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
Find the derivative of x2 cosx.
Find the derivative of f(x) = tan(ax + b), by first principle.
