Advertisements
Advertisements
Question
\[\frac{2x - 1}{x^2 + 1}\]
Advertisements
Solution
\[\text{ Let u } = 2x - 1; v = x^2 + 1; \]
\[\text{ Then }, u' = 2; v' = 2x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{2x - 1}{x^2 + 1} \right) = \frac{\left( x^2 + 1 \right)2 - \left( 2x - 1 \right)2x}{( x^2 + 1 )^2}\]
\[ = \frac{2 x^2 + 2 - 4 x^2 + 2x}{( x^2 + 1 )^2}\]
\[ = \frac{- 2 x^2 + 2x + 2}{( x^2 + 1 )^2}\]
\[ = \frac{2\left( 1 + x - x^2 \right)}{( x^2 + 1 )^2}\]
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):
`(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):
cosec x cot 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):
`(sin x + cos x)/(sin x - 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):
`(sec x - 1)/(sec x + 1)`
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 =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
\[\frac{1}{x^3}\]
k xn
(x2 + 1) (x − 5)
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
x sin x
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
tan2 x
\[\sin \sqrt{2x}\]
\[\tan \sqrt{x}\]
2 sec x + 3 cot x − 4 tan x
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x2 sin x log x
(1 +x2) cos x
sin2 x
x−3 (5 + 3x)
(ax + b)n (cx + d)n
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
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
`(a + b sin x)/(c + d cos x)`
