If $e^{x} + e^{y} = e^{x y}$ $e^x+e^y=e^(xy)$ Then, dy/dx=?

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If $e^{x} + e^{y} = e^{x y}$ $e^x+e^y=e^(xy)$ Then, dy/dx=?

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Answer 1 · 2018-05-09T03:39:10

Use implicit differentiation.

$e^{x} + e^{y} (\frac{d y}{d x}) = (y + x (\frac{d y}{d x})) e^{x y}$ $e^x + e^y(dy/dx) = (y + x(dy/dx))e^(xy)$

$e^{x} + e^{y} (\frac{d y}{d x}) = y e^{x y} + x e^{x y} (\frac{d y}{d x})$ $e^x + e^y(dy/dx) = ye^(xy) + xe^(xy)(dy/dx)$

$e^{y} (\frac{d y}{d x}) - x e^{x y} (\frac{d y}{d x}) = y e^{x y} - e^{x}$ $e^y(dy/dx) - xe^(xy)(dy/dx) = ye^(xy) - e^x$

$\frac{d y}{d x} = \frac{y e^{x y} - e^{x}}{e^{y} - x e^{x y}}$ $dy/dx = (ye^(xy) - e^x)/(e^y - xe^(xy))$

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If $e^{x} + e^{y} = e^{x y}$ $e^x+e^y=e^(xy)$ Then, dy/dx=?

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