Advertisements
Advertisements
प्रश्न
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
Advertisements
उत्तर
\[\text{ We have,} y^x = e^{y - x} \]
Taking log on both sides,
\[\log y^x = \log e^\left( y - x \right) \]
\[ \Rightarrow x\log y = \left( y - x \right)\log e\]
\[ \Rightarrow x\log y = y - x . . . \left( i \right)\]
Differentiating with respect to x,
\[\frac{d}{dx}\left( x \log y \right) = \frac{d}{dx}\left( y - x \right)\]
\[ \Rightarrow \left[ x\frac{d}{dx}\left( \log y \right) + \log y\frac{d}{dx}\left( x \right) \right] = \frac{dy}{dx} - 1\]
\[ \Rightarrow x\left( \frac{1}{y} \right)\frac{dy}{dx} + \log y\left( 1 \right) = \frac{dy}{dx} - 1\]
\[ \Rightarrow \frac{dy}{dx}\left( \frac{x}{y} - 1 \right) = - 1 - \log y\]
\[ \Rightarrow \frac{dy}{dx}\left( \frac{y}{\left( 1 + \log y \right)y} - 1 \right) = - \left( 1 + \log y \right) \left[ Using \left( i \right) \right] \]
\[ \Rightarrow \frac{dy}{dx}\left[ \frac{1 - 1 - \log y}{\left( 1 + \log y \right)} \right] = - \left( 1 + \log y \right)\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{\left( 1 + \log y \right)^2}{- \log y}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\]
APPEARS IN
संबंधित प्रश्न
Differentiate sin (3x + 5) ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
If \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{2 a^x}{1 - a^{2x}} \right), a > 1, - \infty < x < 0\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
If \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?
If \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?
Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?
If \[x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)\] ,prove that \[\frac{dy}{dx} = \frac{x}{y}\]?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?
Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 - a^2 x^2}, \text{ if }-\frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?
If \[f\left( 1 \right) = 4, f'\left( 1 \right) = 2\] find the value of the derivative of \[\log \left( f\left( e^x \right) \right)\] w.r. to x at the point x = 0 ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .
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 = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
If x = a(1 − cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}\] ?
If \[y = e^{\tan^{- 1} x}\] prove that (1 + x2)y2 + (2x − 1)y1 = 0 ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
