Advertisements
Advertisements
Question
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
Solution
\[\text{ Case } 1: x>0\]
\[\left| x \right| = x . . . \left( 1 \right)\]
\[\frac{d}{dx}\left( \log \left| x \right| \right) = \log x\]
\[ = \frac{1}{x}\]
\[ = \frac{1}{\left| x \right|} (\text{ from } (1))\]
\[Case 2:x<0\]
\[\left| x \right| = - x . . . \left( 2 \right)\]
\[\frac{d}{dx}\left( \log \left| x \right| \right) = \log \left( - x \right)\]
\[ = \frac{1}{- x}\]
\[ = \frac{1}{\left| x \right|} (\text{ from } (2))\]
\[\text{ From case } (1) \text{ and case }(2),\]
\[\frac{d}{dx}\left( \log \left| x \right| \right) = \frac{1}{\left| x \right|}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x–3 (5 + 3x).
Find the derivative of x–4 (3 – 4x–5).
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) (cx + d)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):
`(1 + 1/x)/(1- 1/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):
`(a + bsin x)/(c + dcosx)`
Find the derivative of f (x) = 3x at x = 2
\[\frac{x^2 + 1}{x}\]
\[\frac{x + 2}{3x + 5}\]
k xn
x2 + x + 3
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
x2 ex
tan (2x + 1)
\[\cos \sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
2 sec x + 3 cot x − 4 tan x
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
x3 sin x
xn tan x
sin x cos x
x2 sin x log x
(x sin x + cos x) (x cos x − sin x)
(1 − 2 tan x) (5 + 4 sin x)
(1 +x2) cos x
\[e^x \log \sqrt{x} \tan x\]
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
x5 (3 − 6x−9)
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{x}{\sin^n x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
`(a + b sin x)/(c + d cos x)`
