Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 x^{3} \sqrt{y} - 7 e^{y} \log{\left(x \right)}=37

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{3 x^{3} y'}{2 \sqrt{y}} + 9 x^{2} \sqrt{y} - 7 y' e^{y} \log{\left(x \right)} - \frac{7 e^{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(- 9 x^{3} y + 7 \sqrt{y} e^{y}\right)}{x \left(3 x^{3} - 14 \sqrt{y} e^{y} \log{\left(x \right)}\right)}