Advertisements
Advertisements
Question
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
Options
`3/2`
1
0
–1
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 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):
sinn x
x2 + x + 3
\[\sqrt{2 x^2 + 1}\]
Differentiate of the following from first principle:
eax + b
Differentiate of the following from first principle:
− x
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
tan 2x
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
log3 x + 3 loge x + 2 tan x
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
x3 sin x
sin x cos x
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
(2x2 − 3) sin x
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{x}{1 + \tan x}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
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 of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
Find the derivative of 2x4 + x.
Find the derivative of x2 cosx.
