Advertisements
Advertisements
Question
\[\frac{x}{\sin^n x}\]
Advertisements
Solution
\[\text{ Let } u = x; v = \sin^n x\]
\[\text{ Then }, u' = 1; v' = n \sin^{n - 1} x . \cos x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{x}{\sin^n x} \right) = \frac{\sin^n x . 1 - x n \sin^{n - 1} x . \cos x}{\left( \sin^n x \right)^2}\]
\[ = \frac{\sin^{n - 1} x\left( \sin x - nx . \cos x \right)}{\sin^{2n} x}\]
\[ = \frac{\sin x - nx . \cos x}{\sin^{2n - \left( n - 1 \right)} x}\]
\[ = \frac{\sin x - nx\cos x}{\sin^{n + 1} x}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of 99x at x = 100.
Find the derivative of `2x - 3/4`
Find the derivative of x–4 (3 – 4x–5).
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)/(px^2 + qx + r)`
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):
`a/x^4 = b/x^2 + cos 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):
(ax + b)n
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 f (x) = 3x at x = 2
Find the derivative of f (x) = cos x at x = 0
\[\frac{2}{x}\]
\[\frac{x + 1}{x + 2}\]
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
tan2 x
tan (2x + 1)
tan 2x
\[\sqrt{\tan x}\]
\[\sin \sqrt{2x}\]
(2x2 + 1) (3x + 2)
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
xn tan x
sin x cos x
(ax + b)n (cx + d)n
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{x}{1 + \tan x}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
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 of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
Find the derivative of 2x4 + x.
Find the derivative of x2 cosx.
Find the derivative of f(x) = tan(ax + b), by first principle.
(ax2 + cot x)(p + q cos x)
`(a + b sin x)/(c + d cos x)`
