Advertisements
Advertisements
प्रश्न
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
Advertisements
उत्तर
The given function is
`f(x) = x^100/100 + x^99/99 + ....... + x^2/2 + x + 1`
∴ `d/(dx) f(x) = [(x^100)/100 + (x^99)/99 + .... + (x^2)/2 + x + 1]`
`d/(dx) f(x) = d/(dx)(x^100/100) + d/(dx)(x^99/99) + ... + d/(dx) (x^2/2) + d/(dx)(x) + d/(dx)(1)`
On using theorem `d/(dx)(x^n)` = `nx^(n - 1)`, we obtain
`d/(dx) f(x)` = `(100x^99)/100 + (99^98)/99 + ... + (2x)/2 + 1 + 0`
= x99 + x98 + ..... + x + 1
∴ f'(x) = `x^99 + x^98 + ..... + x + 1`
At x = 0,
f'(0) = 1
At x = 1,
f'(1) = `1^99 + 1^98 + ... + 1 + 1 = [1 + 1 + ... + 1 + 1]_(100 "terms")` = 1 × 100 = 100
Thus, f'(1) = 100 × f'(0)
APPEARS IN
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
Find the derivative of x at x = 1.
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):
sinn 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):
`(sin(x + a))/ cos x`
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = 99x at x = 100
Find the derivative of f (x) = tan x at x = 0
\[\frac{1}{\sqrt{3 - x}}\]
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
x sin x
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
3x + x3 + 33
log3 x + 3 loge x + 2 tan x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
(x sin x + cos x ) (ex + x2 log x)
logx2 x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
(2x2 − 3) sin x
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
