Advertisements
Advertisements
प्रश्न
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
Advertisements
उत्तर
\[f'\left( x \right) = \frac{d}{dx}\left( \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 \right)\]
\[ = \frac{1}{100}\left( 100 x^{99} \right) + \frac{1}{99}\left( 99 x^{98} \right) + . . . + \frac{1}{2}\left( 2x \right) + 1 + 0\]
\[ = x^{99} + x^{98} + . . . + x + 1\]
\[f'\left( 1 \right) = 1^{99} + 1^{98} + . . . + 1 + 1\]
\[ = 99 + 1\]
\[ = 100\]
\[f'\left( 0 \right) = 0 + 0 + . . . + 0 + 1\]
\[ = 1\]
\[RHS = 100 f'\left( 0 \right)\]
\[ = 100\left( 1 \right)\]
\[ = 100\]
\[ = f'\left( 1 \right)\]
\[ = LHS\]
\[ \therefore f'\left( 1 \right) = 100 f'\left( 0 \right)\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
Find the derivative of x–4 (3 – 4x–5).
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):
sin (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):
`(sin x + cos x)/(sin x - cos x)`
k xn
(x2 + 1) (x − 5)
\[\sqrt{2 x^2 + 1}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
e3x
x ex
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
x4 − 2 sin x + 3 cos x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{2 x^2 + 3x + 4}{x}\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
x3 sin x
sin x cos x
(x sin x + cos x) (x cos x − sin x)
(1 − 2 tan x) (5 + 4 sin x)
\[e^x \log \sqrt{x} \tan x\]
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
x5 (3 − 6x−9)
x−3 (5 + 3x)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{x}{1 + \tan x}\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
