Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 8 \sqrt{x} \sqrt{y} - 4 e^{x^{3}} e^{y^{3}}=49

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{4 \sqrt{x} y'}{\sqrt{y}} - 12 x^{2} e^{x^{3}} e^{y^{3}} - 12 y^{2} y' e^{x^{3}} e^{y^{3}} + \frac{4 \sqrt{y}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 3 x^{\frac{5}{2}} \sqrt{y} e^{x^{3} + y^{3}} + y}{3 \sqrt{x} y^{\frac{5}{2}} e^{x^{3} + y^{3}} - x}