Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{5} \sqrt{x} \cos{\left(y^{3} \right)} - 6 y^{3} e^{x^{3}}=49

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 18 \sqrt{5} \sqrt{x} y^{2} y' \sin{\left(y^{3} \right)} - 18 x^{2} y^{3} e^{x^{3}} - 18 y^{2} y' e^{x^{3}} + \frac{3 \sqrt{5} \cos{\left(y^{3} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- x^{\frac{5}{2}} y^{3} e^{x^{3}} + \frac{\sqrt{5} \cos{\left(y^{3} \right)}}{6}}{y^{2} \left(\sqrt{x} e^{x^{3}} + \sqrt{5} x \sin{\left(y^{3} \right)}\right)}