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
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Differentiate tan (x° + 45°) ?
Differentiate \[3^{x^2 + 2x}\] ?
Differentiate (log sin x)2 ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
If `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`
If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?
Differentiate \[\left( x^x \right) \sqrt{x}\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\cot x} + \left( \cot x \right)^{\tan x}\] ?
If \[y = \sin \left( x^x \right)\] prove that \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?
If \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?
If \[f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)\] then find \[\frac{dy}{dx} \text { at } x = 1\] ?
If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?
If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{then} \frac{dy}{dx} =\] ____________.
The derivative of \[\sec^{- 1} \left( \frac{1}{2 x^2 + 1} \right) \text { w . r . t }. \sqrt{1 + 3 x} \text { at } x = - 1/3\]
For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text { at } \left( 1/4, 1/4 \right)\text { is }\] _____________ .
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
Find the second order derivatives of the following function x3 + tan x ?
Find the second order derivatives of the following function ex sin 5x ?
If y = x3 log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?
If y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If y = etan x, then (cos2 x)y2 =
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
Differentiate `log [x+2+sqrt(x^2+4x+1)]`
