Advertisements
Advertisements
Question
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Advertisements
Solution
\[\text{ Let y} = \left( \sin x \right)^{\cos x }. . . \left( i \right)\]
Taking log on both sides
\[\log y = \log \left( \sin x \right)^{\cos x }\]
\[ \Rightarrow \log y = \cos x \log \sin x \]
\[\text{ Differentiating with respect to x }, \]
\[\frac{1}{y}\frac{dy}{dx} = \cos x\frac{d}{dx}\left( \log \sin x \right) + \log \sin x\frac{d}{dx}\left( \cos x \right) \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \cos x\frac{1}{\sin x}\frac{d}{dx}\left( \sin x \right) + \log \sin x\left( - \sin x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{\cos x}{\sin x}\left( \cos x \right) - \sin x \log \sin x\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \cos x \cot x - \sin x \log \sin x \right]\]
\[ \Rightarrow \frac{dy}{dx} = \left( \sin x \right)^{\cos x} \left[ \cos x \cot x - \sin x \log \sin x \right] \left[ \text{using equation }\left( i \right) \right]\]
APPEARS IN
RELATED QUESTIONS
If the function f(x)=2x3−9mx2+12m2x+1, where m>0 attains its maximum and minimum at p and q respectively such that p2=q, then find the value of m.
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[e^x \log \sin 2x\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
If \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[x^{\cos^{- 1} x}\] ?
If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?
If \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} \text{ at }\theta = \frac{\pi}{3} \] ?
Differentiate \[\tan^{- 1} \left( \frac{1 + ax}{1 - ax} \right)\] with respect to \[\sqrt{1 + a^2 x^2}\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
If \[x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =\] ____________ .
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\] then `f' (x)` is equal to ____________ .
If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .
If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?
\[ \text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2} \] ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
If x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
