Advertisements
Advertisements
Question
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
Options
`3/2`
1
0
–1
Advertisements
Solution
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is 1.
Explanation:
Given f(x) = x – [x]
We have to first check for differentiability of f(x) at x = `1/2`
∴ Lf'`(1/2)` = L.H.D
= `lim_(h -> 0) (f[1/2 - h] - f[1/2])/(-h)`
= `lim_(h -> 0) ((1/2 - h) - [1/2 - h] - 1/2 + [1/2])/(-h)`
= `lim_(h -> 0) (1/2 - h - 0 - 1/2 + 0)/(-h)`
= `(-h)/(-h)`
= 1
Rf'`(1/2)` = R.H.D
= `lim_(h -> 0) (f(1/2 + h) - f(1/2))/h`
= `lim_(h -> 0) ((1/2 + h) - [1/2 + h] - 1/2 + [1/2])/h`
= `lim_(h -> 0) (1/2 + h - 1 - 1/2 + 1)/h`
= `h/h`
= 1
Since L.H.D = R.H.D
∴ f'`(1/2)` = 1
APPEARS IN
RELATED QUESTIONS
Find the derivative of `2x - 3/4`
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):
`(ax + b)/(cx + d)`
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):
sin (x + a)
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
\[\frac{1}{x^3}\]
(x2 + 1) (x − 5)
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
x4 − 2 sin x + 3 cos x
log3 x + 3 loge x + 2 tan x
\[\frac{2 x^2 + 3x + 4}{x}\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
(x3 + x2 + 1) sin x
(x sin x + cos x) (x cos x − sin x)
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x^5 - \cos x}{\sin x}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
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
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of 2x4 + x.
`(a + b sin x)/(c + d cos x)`
