Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 3 x^{3} e^{y} - 4 y^{3} \cos{\left(x \right)}=39$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 3 x^{3} y' e^{y} + 9 x^{2} e^{y} + 4 y^{3} \sin{\left(x \right)} - 12 y^{2} y' \cos{\left(x \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{9 x^{2} e^{y} + 4 y^{3} \sin{\left(x \right)}}{3 x^{3} e^{y} - 12 y^{2} \cos{\left(x \right)}}$