Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 15 \sqrt{x} \cos{\left(y^{2} \right)} + 2 x^{3} e^{y^{3}}=38

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 30 \sqrt{x} y y' \sin{\left(y^{2} \right)} + 6 x^{3} y^{2} y' e^{y^{3}} + 6 x^{2} e^{y^{3}} + \frac{15 \cos{\left(y^{2} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{x^{\frac{5}{2}} e^{y^{3}} + \frac{5 \cos{\left(y^{2} \right)}}{4}}{y \left(- x^{\frac{7}{2}} y e^{y^{3}} + 5 x \sin{\left(y^{2} \right)}\right)}