Advertisements
Advertisements
प्रश्न
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
पर्याय
150
−50
−150
50
उत्तर
\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]
Differentiating both sides with respect to x, we get
\[f'\left( x \right) = \frac{d}{dx}\left( 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100} \right)\]
\[ = \frac{d}{dx}\left( 1 \right) - \frac{d}{dx}\left( x \right) + \frac{d}{dx}\left( x^2 \right) - \frac{d}{dx}\left( x^3 \right) + . . . - \frac{d}{dx}\left( x^{99} \right) + \frac{d}{dx}\left( x^{100} \right)\]
\[ = 0 - 1 + 2x - 3 x^2 + . . . - 99 x^{98} + 100 x^{99} \]
\[ = - 1 + 2x - 3 x^2 + . . . - 99 x^{98} + 100 x^{99}\]
Putting x = 1, we get
\[f'\left( 1 \right) = - 1 + 2 - 3 + . . . - 99 + 100\]
\[ = \left( - 1 + 2 \right) + \left( - 3 + 4 \right) + \left( - 5 + 6 \right) + . . . + \left( - 99 + 100 \right)\]
\[ = 1 + 1 + 1 + . . . + 1 \left( 50 \text{ terms } \right)\]
\[ = 50\]
Hence, the correct answer is option (d).
APPEARS IN
संबंधित प्रश्न
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):
`(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):
`(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):
`(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):
sinn x
\[\frac{x^2 + 1}{x}\]
\[\frac{x + 1}{x + 2}\]
\[\frac{1}{\sqrt{3 - x}}\]
\[\sqrt{2 x^2 + 1}\]
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
x4 − 2 sin x + 3 cos x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
x2 sin x log x
(x sin x + cos x) (x cos x − sin x)
x−3 (5 + 3x)
(ax + b) (a + d)2
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{x + \cos x}{\tan x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
If f (x) = \[\log_{x_2}\]write the value of f' (x).
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 of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
(ax2 + cot x)(p + q cos x)
`(a + b sin x)/(c + d cos x)`
