Advertisements
Advertisements
प्रश्न
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
Advertisements
उत्तर
The given series is a geometric series where a = 1 and r = x.
\[f\left( x \right) = 1 + x + x^2 + x^3 + . . . = \frac{1}{1 - x}\]
\[\left( \text{ Sum of the infinite series of a geometric series is }\frac{a}{1 - r}. \right)\]
\[f'\left( x \right) = \frac{- 1}{(1 - x )^2}\frac{d}{dx}(1 - x)\]
\[ = \frac{- 1}{(1 - x )^2}( - 1)\]
\[ = \frac{1}{(1 - x )^2}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
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)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):
(ax + b)n (cx + d)m
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
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{1}{\sqrt{x}}\]
(x2 + 1) (x − 5)
tan 2x
3x + x3 + 33
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x3 ex
(1 − 2 tan x) (5 + 4 sin x)
x5 (3 − 6x−9)
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
(ax + b) (a + d)2
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{x}{\sin^n x}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
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 \[f'\left( a \right)\]
`(a + b sin x)/(c + d cos x)`
