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 x2 – 2 at x = 10.
Find the derivative of 99x at x = 100.
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
Find the derivative of (5x3 + 3x – 1) (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):
`(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):
`(px^2 +qx + r)/(ax +b)`
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
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
k xn
Differentiate of the following from first principle:
− x
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
2 sec x + 3 cot x − 4 tan x
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
sin x cos x
(x sin x + cos x ) (ex + x2 log x)
x4 (5 sin x − 3 cos x)
(2x2 − 3) sin x
x−3 (5 + 3x)
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
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
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.
