Advertisements
Advertisements
प्रश्न
\[\frac{x}{\sin^n x}\]
Advertisements
उत्तर
\[\text{ Let } u = x; v = \sin^n x\]
\[\text{ Then }, u' = 1; v' = n \sin^{n - 1} x . \cos x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{x}{\sin^n x} \right) = \frac{\sin^n x . 1 - x n \sin^{n - 1} x . \cos x}{\left( \sin^n x \right)^2}\]
\[ = \frac{\sin^{n - 1} x\left( \sin x - nx . \cos x \right)}{\sin^{2n} x}\]
\[ = \frac{\sin x - nx . \cos x}{\sin^{2n - \left( n - 1 \right)} x}\]
\[ = \frac{\sin x - nx\cos x}{\sin^{n + 1} x}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x–3 (5 + 3x).
Find the derivative of x5 (3 – 6x–9).
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):
(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):
`(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)`
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):
`4sqrtx - 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):
`cos x/(1 + sin x)`
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan2 x
\[\sqrt{\tan x}\]
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
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
logx2 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)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
`(a + b sin x)/(c + d cos x)`
