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 x2 – 2 at x = 10.
Find the derivative of x at x = 1.
Find the derivative of x–3 (5 + 3x).
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)`
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 + 1/x)/(1- 1/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):
`a/x^4 = b/x^2 + cos 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):
`(a + bsin x)/(c + dcosx)`
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 f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = tan x at x = 0
\[\frac{x^2 + 1}{x}\]
\[\frac{x + 1}{x + 2}\]
\[\frac{2x + 3}{x - 2}\]
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:
x2 sin x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
xn loga x
x4 (5 sin x − 3 cos x)
x−3 (5 + 3x)
\[\frac{e^x}{1 + x^2}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{3^x}{x + \tan x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of 2x4 + x.
Find the derivative of f(x) = tan(ax + b), by first principle.
