Solve the following : If e ^ x + e ^ y = e ^ ( x + y ) , then sho… - Vidyadip

Question

Solve the following : If $e ^{ x }+ e ^{ y }= e ^{( x + y )}$, then show that $\frac{d y}{d x}=-e^{y-x}$.

✓

Answer

$
e ^{ x }+ e ^{ y }= e ^{( x + y )}
$
Differentiating both sides w.r.t. $x$, we get
$
\begin{aligned}
& e^x+e^y \cdot \frac{d y}{d x}=e^{(x+y)} \cdot \frac{d}{d x}(x+y) \\
& \therefore e^x+e^y \cdot \frac{d y}{d x}=e^{(x+y)} \cdot\left(1+\frac{d y}{d x}\right) \\
& \therefore e^x+e^y \cdot \frac{d y}{d x}=e^{(x+y)}+e^{(x+y)} \cdot \frac{d y}{d x}
\end{aligned}
$
$
\begin{aligned}
& \therefore\left[e^y-e^{(x+y)}\right] \frac{d y}{d x}=e^{(x+y)}-e^x \\
& \therefore\left(e^y-e^x-e^y\right) \frac{d y}{d x}=e^x+e^y-e^x \\
& \therefore-e^x \cdot \frac{d y}{d x}=e^y \\
& \therefore \frac{d y}{d x}=-\frac{e^y}{e^x}=-e^{y-x} .
\end{aligned}
$

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