Advertisements
Advertisements
Question
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
Advertisements
Solution
\[\text{ Let } u = 4x + 5 \sin x; v = 3x + 7 \cos x\]
\[\text{ Then }, u' = 4 + 5 \cos x; v' = 3 - 7 \sin x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{4x + 5 \sin x}{3x + 7 \cos x} \right) = \frac{\left( 3x + 7 \cos x \right)\left( 4 + 5 \cos x \right) - \left( 4x + 5 \sin x \right)\left( 3 - 7 \sin x \right)}{\left( 3x + 7 \cos x \right)^2}\]
\[ = \frac{12x + 15 x \cos x + 28 \cos x + 35 \cos^2 x - 12x + 28 x \sin x - 15 \sin x + 35 \sin^2 x}{\left( 3x + 7 \cos x \right)^2}\]
\[ = \frac{15 x \cos x + 28 x \sin x + 28 \cos x15 \sin x + 35\left( \sin^2 x + \cos^2 x \right)}{\left( 3x + 7 \cos x \right)^2}\]
\[ = \frac{15 x \cos x + 28 x \sin x + 28 \cos x15 \sin x + 35}{\left( 3x + 7 \cos x \right)^2}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of (5x3 + 3x – 1) (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):
`cos x/(1 + sin x)`
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
\[\frac{1}{x^3}\]
\[\frac{x^2 + 1}{x}\]
\[\frac{x^2 - 1}{x}\]
x2 + x + 3
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
sin (x + 1)
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
x cos x
\[\tan \sqrt{x}\]
\[\tan \sqrt{x}\]
ex log a + ea long x + ea log a
log3 x + 3 loge x + 2 tan x
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
2 sec x + 3 cot x − 4 tan x
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
x3 ex
sin x cos x
\[e^x \log \sqrt{x} \tan x\]
x3 ex cos x
x−3 (5 + 3x)
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{x + \cos x}{\tan x}\]
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\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
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
