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 \[f'\left( 1 \right)\]
विकल्प
150
−50
−150
50
Advertisements
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 x–3 (5 + 3x).
Find the derivative of x–4 (3 – 4x–5).
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 + 1/x)/(1- 1/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):
sinn x
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
eax + b
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x3 sin x
xn tan x
x5 ex + x6 log x
(x sin x + cos x ) (ex + x2 log x)
(1 − 2 tan x) (5 + 4 sin x)
x4 (5 sin x − 3 cos x)
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
(ax + b) (a + d)2
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in 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
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 of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Find the derivative of 2x4 + x.
Find the derivative of f(x) = tan(ax + b), by first principle.
