Advertisements
Advertisements
प्रश्न
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
उत्तर
\[\text{ Let y } = \left( \cos x \right)^x + \left( \sin x \right)^\frac{1}{x} \]
\[ \Rightarrow y = e^{ \log \left( \cos x\right)^x} + e^{\log \left( \sin x \right)^\frac{1}{x} } \]
\[ \Rightarrow y = e^{ x\log\left( \cos x \right) } + e^\frac{1}{x}\log\sin x\]
Differentiating with respect to x,
\[\frac{dy}{dx} = \frac{d}{dx}\left( e^{x \log\cos x} \right) + \frac{d}{dx}\left( e^\frac{1}{x}\log \sin x \right)\]
\[ = e^{x \log\cos x} \times \frac{d}{dx}\left( x \log\cos x \right) + e^\frac{1}{x}\log \sin x \frac{d}{dx}\left( \frac{1}{x}\log\sin x \right)\]
\[ = e^{\log \left( \cos x \right)^x }\times \left[ x\frac{d}{dx}\left( \log\cos x \right) + \log\cos x \times \frac{d}{dx}\left( x \right) \right] + e^{\log \left( \sin x \right)^\frac{1}{x} }\times \left[ \frac{1}{x}\frac{d}{dx}\left( \log\sin x \right) + \log\sin x\frac{d}{dx}\left( \frac{1}{x} \right) \right]\]
\[ = \left( \cos x \right)^x \left[ x\left( \frac{1}{\cos x} \right)\frac{d}{dx}\left( \cos x \right) + \log\cos x\left( 1 \right) \right] + \left( \sin \right)^\frac{1}{x} \left[ \frac{1}{x} \times \frac{1}{\sin x} \times \frac{d}{dx}\left( \sin x \right) + \log\sin x\left( - \frac{1}{x^2} \right) \right]\]
\[ = \left( \cos x \right)^x \left[ x\left( \frac{1}{\cos x} \right)\left( - \sin x \right) + \log\cos x \right] + \left( \sin x \right)^\frac{1}{x} \left[ \frac{1}{x} \times \frac{1}{\sin x}\left( \cos x \right) - \frac{1}{x^2}\log\sin x \right]\]
\[ = \left( \cos x \right)^x \left[ \log\cos x - x \tan x \right] + \left( \sin x \right)^\frac{1}{x} \left[ \frac{\cot x}{x} - \frac{1}{x^2}\log\sin x \right]\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles eax+b.
Differentiate \[3^{x \log x}\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\], prove that \[2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}\] ?
If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
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 \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?
If \[\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[e^{x \log x}\] ?
If \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] when \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?
Find \[\frac{dy}{dx}\] , when \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?
If \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?
Differentiate x2 with respect to x3
Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { if } 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text{ if } - 1 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 - a^2 x^2}, \text{ if }-\frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?
If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
If y = e−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?
If y = x3 log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
If y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
Differentiate `log [x+2+sqrt(x^2+4x+1)]`
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
