If x^3+y^2+x y=7, find d y d x - Vidyadip

Question

If $x^3+y^2+x y=7$, find $\frac{d y}{d x}$

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Answer

$
x^3+y^2+x y=7
$
Differentiating both sides w.r.t. $x$, we get
$
\begin{aligned}
& 3 x^2+2 y \cdot \frac{d y}{d x}+x \frac{d y}{d x}+y \cdot \frac{d}{d x}(x)=0 \\
& \therefore 3 x^2+2 y \frac{d y}{d x}+x \frac{d y}{d x}+y \times 1=0
\end{aligned}
$
$
\begin{aligned}
& \therefore(2 y+x) \frac{d y}{d x}=-3 x^2-y \\
& \therefore \frac{d y}{d x}=\frac{-\left(y+3 x^2\right)}{2 y+x} .
\end{aligned}
$

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