Advertisements
Advertisements
प्रश्न
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
Advertisements
उत्तर
\[\text{ Given that }y^x + x^y + x^x = a^b \]
\[\text{ Putting u }= y^x , v = x^y \text{and }w = x^x , \text{ we get }\]
\[ u + v + w = a^b \]
\[ \therefore \frac{du}{dx} + \frac{dv}{dx} + \frac{dw}{dx} = 0 . . . \left( i \right)\]
\[\text{ Now, u } = y^x \]
Taking log on both sides,
\[\log u = x \log y\]
\[\Rightarrow \frac{1}{u}\frac{du}{dx} = x\frac{d}{dx}\left( \log y \right) + \log y\frac{d}{dx}\left( x \right) \left[ \text{ using product } rule \right]\]
\[ \Rightarrow \frac{1}{u}\frac{du}{dx} = x\frac{1}{y}\frac{dy}{dx} + \log y \times 1\]
\[ \Rightarrow \frac{du}{dx} = u\left( \frac{x}{y}\frac{dy}{dx} + \log y \right)\]
\[ \Rightarrow \frac{du}{dx} = y^x \left( \frac{x}{y}\frac{dy}{dx} + \log y \right) . . . \left( ii \right)\]
\[\text{ Also, v } = x^y\]
Taking log on both sides,
\[\log v = y \log x\]
\[\Rightarrow \frac{1}{v}\frac{dv}{dx} = y\frac{d}{dx}\left( \log x \right) + \log x\frac{dy}{dx}\]
\[ \Rightarrow \frac{1}{v}\frac{dv}{dx} = y\frac{1}{x} + \log x\frac{dy}{dx}\]
\[ \Rightarrow \frac{dv}{dx} = v\left[ \frac{y}{x} + \log x\frac{dy}{dx} \right]\]
\[ \Rightarrow \frac{dv}{dx} = x^y \left[ \frac{y}{x} + \log x\frac{dy}{dx} \right] . . . \left( iii \right)\]
\[\text{ Again, w } = x^x\]
Taking log on both sides,
\[\log w = x \log x\]
\[\Rightarrow \frac{1}{w}\frac{dw}{dx} = x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx}\left( x \right)\]
\[ \Rightarrow \frac{1}{w}\frac{dw}{dx} = x\frac{1}{x} + \log x\left( 1 \right)\]
\[ \Rightarrow \frac{dw}{dx} = w\left( 1 + \log x \right)\]
\[ \Rightarrow \frac{dw}{dx} = x^x \left( 1 + \log x \right) . . . \left( iv \right)\]
\[\text{ From } \left( i \right), \left( ii \right), \left( iii \right)\text{ and }\left( iv \right), \text{ we have }\]
\[ y^x \left( \frac{x}{y}\frac{dy}{dx} + \log y \right) + x^y \left( \frac{y}{x} + \log x\frac{dy}{dx} \right) + x^x \left( 1 + \log x \right) = 0\]
\[ \Rightarrow \left( x . y^{x - 1} + x^y . \log x \right)\frac{dy}{dx} = - x^x \left( 1 + \log x \right) - y . x^{y - 1} - y^x \log y\]
\[ \therefore \frac{dy}{dx} = \frac{- \left\{ y^x \log y + y . x^{y - 1} + x^x \left( 1 + \log x \right) \right\}}{x . y^{x - 1} + x^y \log x}\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
Differentiate
\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[x^{\cos^{- 1} x}\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
If \[x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}\] ?
If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
If \[y = \frac{\log x}{x}\] show that \[\frac{d^2 y}{d x^2} = \frac{2 \log x - 3}{x^3}\] ?
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 (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = ae2x + be−x, show that, \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\] ?
If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
