Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 9 x^{3} e^{y} + 7 y^{3} \log{\left(x \right)}=40$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 9 x^{3} y' e^{y} + 27 x^{2} e^{y} + 21 y^{2} y' \log{\left(x \right)} + \frac{7 y^{3}}{x} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{27 x^{3} e^{y} + 7 y^{3}}{9 x^{4} e^{y} + 21 x y^{2} \log{\left(x \right)}}$