Advertisements
Advertisements
प्रश्न
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
विकल्प
5050
5049
5051
50051
Advertisements
उत्तर
\[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\]
Differentiating both sides with respect to x, we get \[f'\left( x \right) = \frac{d}{dx}\left( x^{100} + x^{99} + . . . + x + 1 \right)\]
\[ = \frac{d}{dx}\left( x^{100} \right) + \frac{d}{dx}\left( x^{99} \right) + . . . + \frac{d}{dx}\left( x^2 \right) + \frac{d}{dx}\left( x \right) + \frac{d}{dx}\left( 1 \right)\]
\[ = 100 x^{99} + 99 x^{98} + . . . + 2x + 1 + 0 \left( y = x^n \Rightarrow \frac{dy}{dx} = n x^{n - 1} \right)\]
\[ = 100 x^{99} + 99 x^{98} + . . . + 2x + 1\]
Putting x = 1, we get
\[f'\left( 1 \right) = 100 + 99 + 98 + . . . + 2 + 1\]
\[ = \frac{100\left( 100 + 1 \right)}{2} \left( S_n = \frac{n\left( n + 1 \right)}{2} \right)\]
\[ = 50 \times 101\]
\[ = 5050\]
Hence, the correct answer is option (a).
APPEARS IN
संबंधित प्रश्न
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):
`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)/(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):
`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):
`4sqrtx - 2`
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):
`cos x/(1 + sin 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):
`(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):
x4 (5 sin x – 3 cos x)
Find the derivative of f (x) = 99x at x = 100
Find the derivative of f (x) = cos x at x = 0
\[\frac{1}{\sqrt{x}}\]
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
\[3^{x^2}\]
tan 2x
\[\tan \sqrt{x}\]
log3 x + 3 loge x + 2 tan x
2 sec x + 3 cot x − 4 tan x
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x3 ex
(x3 + x2 + 1) sin x
sin x cos x
\[\frac{2^x \cot x}{\sqrt{x}}\]
x5 ex + x6 log x
(ax + b) (a + d)2
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
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 of the following:
If\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
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 each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
