Advertisements
Advertisements
प्रश्न
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
विकल्प
`3/2`
1
0
–1
उत्तर
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
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
Find the derivative of x5 (3 – 6x–9).
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/(ax^2 + bx + c)`
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):
x4 (5 sin x – 3 cos x)
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
k xn
Differentiate of the following from first principle:
eax + b
x ex
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
\[3^{x^2}\]
tan 2x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
2 sec x + 3 cot x − 4 tan x
sin x cos x
(1 +x2) cos x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
(ax + b) (a + d)2
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
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 of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
Find the derivative of 2x4 + x.
