Advertisements
Advertisements
प्रश्न
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
Advertisements
उत्तर
\[\text{ Let } u = a x^2 + bx + c; v = p x^2 + qx + r\]
\[\text{ Then }, u' = 2ax + b; v' = 2px + q\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{a x^2 + bx + c}{p x^2 + qx + r} \right) = \frac{\left( p x^2 + qx + r \right)\left( 2ax + b \right) - \left( a x^2 + bx + c \right)\left( 2px + q \right)}{\left( p x^2 + qx + r \right)^2}\]
\[ = \frac{2ap x^3 + 2aq x^2 + 2arx + bp x^2 + bqx + br - 2ap x^3 - 2bp x^2 - 2pcx - aq x^2 - bqx - cq}{\left( p x^2 + qx + r \right)^2}\]
\[ = \frac{\left( aq - bp \right) x^2 + 2\left( ar - xp \right)x + br - cq}{\left( p x^2 + qx + r \right)^2}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
Find the derivative of `2x - 3/4`
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):
`cos x/(1 + sin 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 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 + bsin x)/(c + dcosx)`
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):
x4 (5 sin x – 3 cos x)
Find the derivative of f (x) = 99x at x = 100
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
\[\frac{x + 2}{3x + 5}\]
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
− x
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
tan (2x + 1)
\[\sqrt{\tan x}\]
(2x2 + 1) (3x + 2)
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x2 sin x log x
(x sin x + cos x) (x cos x − sin x)
logx2 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.
(3x2 + 2)2
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)
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Find the derivative of x2 cosx.
Find the derivative of f(x) = tan(ax + b), by first principle.
