Advertisements
Advertisements
Question
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
Advertisements
Solution
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
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
Find the derivative of `2x - 3/4`
Find the derivative of (5x3 + 3x – 1) (x – 1).
Find the derivative of x–4 (3 – 4x–5).
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):
(x + a)
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)/(px^2 + qx + r)`
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 f (x) = tan x at x = 0
\[\frac{1}{\sqrt{3 - x}}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
tan (2x + 1)
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
(x3 + x2 + 1) sin x
x2 sin x log x
x5 ex + x6 log x
(x sin x + cos x ) (ex + x2 log x)
(1 − 2 tan x) (5 + 4 sin x)
logx2 x
(2x2 − 3) sin x
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{x}{\sin^n x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
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 x2 cosx.
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
