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 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):
`(px+ q) (r/s + s)`
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):
sin (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):
`(sec x - 1)/(sec x + 1)`
\[\frac{2}{x}\]
\[\frac{1}{\sqrt{x}}\]
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
sin (x + 1)
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
\[\tan \sqrt{x}\]
3x + x3 + 33
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
(2x2 + 1) (3x + 2)
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
cos (x + a)
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
sin x cos x
(x sin x + cos x ) (ex + x2 log x)
sin2 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.
(x + 2) (x + 3)
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{x}{\sin^n x}\]
\[\frac{1}{a x^2 + bx + c}\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
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
