Find d y d x if, : y e^x+x e^y=1 - Vidyadip

Question

Find $\frac{d y}{d x}$ if, :
$
y \cdot e^x+x \cdot e^y=1
$

✓

Answer

$
y \cdot e^x+x \cdot e^y=1
$
Differentiating both sides w.r.t. $x_r$ we get
$
\begin{aligned}
& \frac{d}{d x}\left(y e^x\right)+\frac{d}{d x}\left(x e^y\right)=0 \\
& \therefore y \cdot \frac{d}{d x}\left(e^x\right)+e^x \cdot \frac{d y}{d x}+x \cdot \frac{d}{d x}\left(e^y\right)+e^y \cdot \frac{d}{d x}(x)=0 \\
& \therefore y \cdot e^x+e^x \cdot \frac{d y}{d x}+x \cdot e^y \cdot \frac{d y}{d x}+e^y \times 1=0 \\
& \therefore\left(e^x+x e^y\right) \frac{d y}{d x}=-e^y-y e^x \\
& \therefore \frac{d y}{d x}=-\left(\frac{e^y+y e^x}{e^x+x e^y}\right) .
\end{aligned}
$

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