xy=e^x y (Differentiating both sides) y+xyy^‵ #160;=e^x y(1 y^‵) y+xyy^‵ =e^x y e^x y× y^‵ y^‵ (xy+e^x y)=e^x y y y^‵ =e^x y yxy+e^x y ...

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Implicit Differentiation

Find dydx i...

Question

Find $\frac{d y}{d x}$ if $x y = e^{x - y}$

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Solution

$x y = e^{x - y}$ (Differentiating both sides)
$y + x y y^{‵} = e^{x - y} (1 - y^{‵}) y + x y y^{‵} = e^{x - y} - e^{x - y} \times y^{‵} y^{‵} (x y + e^{x - y}) = e^{x - y} - y y^{‵} = \frac{e^{x - y} - y}{x y + e^{x - y}}$

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