[tex]x^{y} =e^{x-y}[/tex]
find [tex]\frac{dy}{dx}[/tex]
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[tex]x^{y} =e^{x-y}[/tex]
find [tex]\frac{dy}{dx}[/tex]
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[tex]\large\underline{\sf{Solution-}}[/tex]
Given function is
[tex]\sf \: {x}^{y} = {e}^{x - y} \\ \\ [/tex]
On taking log on both sides, we get
[tex]\sf \: log{x}^{y} = log[ {e}^{x - y} ] \\ \\ [/tex]
[tex]\sf \: y \: logx \: = \: (x - y) \: loge \\ \\ [/tex]
[tex]\sf \: y \: logx \: = \: x - y \: \: \: \: \: \: \: \: \: \: \{ \because \: loge = 1 \} \\ \\ [/tex]
[tex]\sf \: y \: logx + y\: = \: x \\ \\ [/tex]
[tex]\sf \: y( logx + 1)\: = \: x \\ \\ [/tex]
[tex]\sf \: y = \dfrac{x}{logx + 1} \\ \\ [/tex]
On differentiating both sides w. r. t. x, we get
[tex]\sf \:\dfrac{d}{dx} y = \dfrac{d}{dx}\bigg(\dfrac{x}{logx + 1}\bigg) \\ \\ [/tex]
[tex]\sf \: \dfrac{dy}{dx} = \dfrac{(logx + 1)\dfrac{d}{dx}x - x\dfrac{d}{dx}(logx + 1)}{ {(logx + 1)}^{2} } \\ \\ [/tex]
[tex]\sf \: \dfrac{dy}{dx} = \dfrac{(logx + 1) \times 1 - x \times \dfrac{1}{x} }{ {(logx + 1)}^{2} } \\ \\ [/tex]
[tex]\sf \: \dfrac{dy}{dx} = \dfrac{logx + 1 - 1 }{ {(logx + 1)}^{2} } \\ \\ [/tex]
[tex]\sf \: \bf\implies \:\dfrac{dy}{dx} = \dfrac{logx}{ {(logx + 1)}^{2} } \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Formulae Used :-
[tex]\boxed{ \sf{ \:log( {x}^{y} ) = y \: logx \: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\dfrac{d}{dx} \frac{u}{v} \: = \: \frac{v\dfrac{d}{dx}u \: - \: u\dfrac{d}{dx}v}{ \: \: {v}^{2} \: \: t } \: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\dfrac{d}{dx}logx = \frac{1}{x} \: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\dfrac{d}{dx}x = 1 \: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\dfrac{d}{dx}k = 0\: }} \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {x}^{n} & \sf {nx}^{n - 1}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}[/tex]