Advertisements
Advertisements
प्रश्न
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to
पर्याय
\[\frac{1}{100}\]
100
50
0
Advertisements
उत्तर
\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\]
Differentiating both sides with respect to x, we get
\[f'\left( x \right) = \frac{d}{dx}\left( 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100} \right)\]
\[ = \frac{d}{dx}\left( 1 \right) + \frac{d}{dx}\left( x \right) + \frac{d}{dx}\left( \frac{x^2}{2} \right) + . . . + \frac{d}{dx}\left( \frac{x^{100}}{100} \right)\]
\[ = \frac{d}{dx}\left( 1 \right) + \frac{d}{dx}\left( x \right) + \frac{1}{2}\frac{d}{dx}\left( x^2 \right) + . . . + \frac{1}{100}\frac{d}{dx}\left( x^{100} \right)\]
\[ = 0 + 1 + \frac{1}{2} \times 2x + . . . + \frac{1}{100} \times 100 x^{99} \left( y = x^n \Rightarrow \frac{dy}{dx} = n x^{n - 1} \right) \]
\[ = 1 + x + x^2 + . . . + x^{99}\]
Putting x = 1, we get
\[f'\left( 1 \right) = 1 + 1 + 1 + . . . + 1 \left( 100 \text{ terms } \right)\]
\[ = 100\]
Hence, the correct answer is option (b).
APPEARS IN
संबंधित प्रश्न
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):
`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):
(ax + b)n (cx + d)m
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):
`(sec x - 1)/(sec x + 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))/ cos x`
Find the derivative of f (x) = 99x at x = 100
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
\[\frac{x^2 - 1}{x}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
\[\sqrt{\tan x}\]
\[\sin \sqrt{2x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
2 sec x + 3 cot x − 4 tan x
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
cos (x + a)
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x2 ex log x
sin x cos x
x2 sin x log x
(1 − 2 tan x) (5 + 4 sin x)
(ax + b)n (cx + d)n
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
Find the derivative of f(x) = tan(ax + b), by first principle.
