Advertisements
Advertisements
Question
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
Options
\[\frac{1}{100}\]
100
50
0
Advertisements
Solution
\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\]
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
RELATED QUESTIONS
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):
(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):
`(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):
`(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):
`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):
cosec x cot x
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) = 99x at x = 100
Find the derivative of f (x) x at x = 1
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
\[\frac{x + 2}{3x + 5}\]
Differentiate each of the following from first principle:
e−x
x ex
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
tan 2x
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Find the derivative of x2 cosx.
