Advertisements
Advertisements
प्रश्न
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
उत्तर
\[f\left( x \right) = \left| x \right| + \left| x - 1 \right|\]
\[\text{ Case }1: x<0 (\therefore x-1<-1<0)\]
\[\left| x \right| = - x; \left| x - 1 \right| = - \left( x - 1 \right) = - x + 1\]
\[f\left( x \right) = - x + \left( - x + 1 \right) = - 2x\]
\[f'\left( x \right) = - 2\]
\[\text{ Case } 2: 0< x <1 (\therefore x>0 \text{ and } x-1<0)\]
\[\left| x \right| = x; \left| x - 1 \right| = - \left( x - 1 \right) = 1 - x\]
\[f\left( x \right) = x + 1 - x = 1\]
\[f'\left( x \right) = 0\]
\[\text{ Case } 3: x>1 \therefore x>1>0 \Rightarrow x>0)\]
\[\left| x \right| = x; \left| x - 1 \right| = x - 1\]
\[f\left( x \right) = x + x - 1 = 2x - 1\]
\[f'\left( x \right) = 2\]
\[f'(x)=\begin{cases}-2, \text{When } x < 0 \\0, \text{When }0 < x <1\\2, \text{When } x >1 \end{cases}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
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):
`(px+ q) (r/s + s)`
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):
`(ax + b)/(px^2 + qx + r)`
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):
`(sec x - 1)/(sec x + 1)`
Find the derivative of f (x) = 99x at x = 100
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
x ex
Differentiate of the following from first principle:
x cos x
tan 2x
\[\tan \sqrt{x}\]
ex log a + ea long x + ea log a
cos (x + a)
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x3 sin x
x3 ex cos 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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x^5 - \cos x}{\sin x}\]
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 \[\frac{dy}{dx} =\]
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.
`(a + b sin x)/(c + d cos x)`
