Advertisements
Advertisements
Question
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
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 99x at x = 100.
Find the derivative of `2/(x + 1) - x^2/(3x -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):
(ax + b)n (cx + d)m
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) x at x = 1
x2 + x + 3
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
x sin x
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
\[\sin \sqrt{2x}\]
\[\tan \sqrt{x}\]
(2x2 + 1) (3x + 2)
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
cos (x + a)
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x3 ex
xn tan x
\[e^x \log \sqrt{x} \tan x\]
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.
(x + 2) (x + 3)
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
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 of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Find the derivative of 2x4 + x.
