Advertisements
Advertisements
Question
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
Advertisements
Solution
\[\text{ Let } u = \sin x - x \cos x; v = x \sin x + \cos x\]
\[\text{ Then }, u' = \cos x + x \sin x - \cos x; v' = x \cos x + \sin x - \sin x\]
\[ = x \sin x = x \cos x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{\sin x - x \cos x}{x \sin x + \cos x} \right) = \frac{\left( x \sin x + \cos x \right)x \sin x - \left( \sin x - x \cos x \right)x \cos x}{\left( x \sin x + \cos x \right)^2}\]
\[ = \frac{x^2 \sin^2 x + x \cos x \sin x - x \cos x \sin x + x^2 \cos^2 x}{\left( x \sin x + \cos x \right)^2}\]
\[ = \frac{x^2 \left( \sin^2 x + \cos^2 x \right)}{\left( x \sin x + \cos x \right)^2}\]
\[ = \frac{x^2}{\left( x \sin x + \cos x \right)^2}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
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):
`(ax + b)/(cx + d)`
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):
`1/(ax^2 + bx + c)`
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
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))/ cos x`
Find the derivative of f (x) = 99x at x = 100
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
\[\frac{x + 1}{x + 2}\]
\[\frac{1}{\sqrt{3 - x}}\]
(x + 2)3
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
\[\sin \sqrt{2x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
ex log a + ea long x + ea log a
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
x2 ex log x
xn loga x
x2 sin x log x
x−3 (5 + 3x)
(ax + b)n (cx + d)n
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of x2 cosx.
