Find dx/dy xy+3e^y=3e

Differentiate using the Product Rule which states that ${\displaystyle \frac{d}{dy}\left[f\left(y\right)g\left(y\right)\right]}$ is ${\displaystyle f\left(y\right)\frac{d}{dy}\left[g\left(y\right)\right]+g\left(y\right)\frac{d}{dy}\left[f\left(y\right)\right]}$ where ${\displaystyle f\left(y\right)=x}$ and ${\displaystyle g\left(y\right)=y}$.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dy}\left[y^n\right]}$ is ${\displaystyle n{y}^{n-1}}$ where ${\displaystyle n=1}$.

Rewrite ${\displaystyle \frac{d}{dy}\left[x\right]}$ as ${\displaystyle x\mathrm{<mo>\prime </mo>}}$.

Multiply ${\displaystyle x}$ by ${\displaystyle 1}$.

Since ${\displaystyle 3}$ is constant with respect to ${\displaystyle y}$, the derivative of ${\displaystyle 3e^y}$ with respect to ${\displaystyle y}$ is ${\displaystyle 3\frac{d}{dy}\left[e^y\right]}$.

Differentiate using the Exponential Rule which states that ${\displaystyle \frac{d}{dy}\left[a^y\right]}$ is ${\displaystyle a^y\ln \left(a\right)}$ where ${\displaystyle a}$=${\displaystyle e}$.

Subtract ${\displaystyle x}$ from both sides of the equation.

Subtract ${\displaystyle 3e^y}$ from both sides of the equation.

Divide each term in ${\displaystyle yx\mathrm{<mo>\prime </mo>}=-x-3e^y}$ by ${\displaystyle y}$.

Simplify the left side.

Simplify the right side.