Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=y$

Applying the derivative of the exponential function

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=y$

Group the terms of the equation by moving the terms that have the variable $y^{\prime}$ to the left side, and those that do not have it to the right side

Move everything to the left hand side of the equation

Factoring by $y+xy^{\prime}$

Break the equation in $2$ factors and set each factor equal to zero, to obtain simpler equations

Solve the equation ($1$)

We need to isolate the dependent variable $y$, we can do that by simultaneously subtracting $y$ from both sides of the equation

Divide both sides of the equation by $x$

Solve the equation ($2$)

This equation $-e^{xy}xy+1=0$ has no solutions in the real plane

The solution of the equation is