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 x5 (3 – 6x–9).
Find the derivative of `2/(x + 1) - x^2/(3x -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):
`(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):
`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):
cosec x cot x
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{1}{\sqrt{x}}\]
\[\frac{x^2 + 1}{x}\]
x2 + x + 3
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
(2x2 + 1) (3x + 2)
2 sec x + 3 cot x − 4 tan x
x2 ex log x
sin x cos x
x2 sin x log x
(1 − 2 tan x) (5 + 4 sin x)
\[e^x \log \sqrt{x} \tan x\]
x3 ex cos x
x4 (5 sin x − 3 cos x)
x−3 (5 + 3x)
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.
(x + 2) (x + 3)
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)
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
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 \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
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
(ax2 + cot x)(p + q cos x)
