Advertisements
Advertisements
Question
\[\frac{a + b \sin x}{c + d \cos x}\]
Advertisements
Solution
\[\text{ Let } u = a + b \sin x; v = c + d \cos x\]
\[\text{ Then }, u' = b \cos x; v' = - d \sin x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{a + b \sin x}{c + d \cos x} \right) = \frac{\left( c + d \cos x \right)b \cos x - \left( a + b \sin x \right)\left( - d \sin x \right)}{\left( c + d \cos x \right)^2}\]
\[ = \frac{bc \cos x + bd \cos^2 x + ad \sin x + bd \sin^2 x}{\left( c + d \cos x \right)^2}\]
\[ = \frac{bc \cos x + ad \sin x + bd \left( \sin^2 x + \cos^2 x \right)}{\left( c + d \cos x \right)^2}\]
\[ = \frac{bc \cos x + ad \sin x + bd}{\left( c + d \cos x \right)^2}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
Find the derivative of 99x at x = 100.
Find the derivative of `2x - 3/4`
Find the derivative of x–3 (5 + 3x).
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):
`(1 + 1/x)/(1- 1/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):
`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):
sinn x
\[\frac{1}{x^3}\]
\[\frac{x^2 + 1}{x}\]
(x + 2)3
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
tan2 x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
ex log a + ea long x + ea log a
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
cos (x + a)
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
x3 sin x
x2 ex log x
xn tan x
sin2 x
(2x2 − 3) sin x
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.
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
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 \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
Find the derivative of 2x4 + x.
