Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 15 \sqrt{x} e^{y^{3}} - 9 \sqrt{y} \sin{\left(x^{2} \right)}=14

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 45 \sqrt{x} y^{2} y' e^{y^{3}} - 18 x \sqrt{y} \cos{\left(x^{2} \right)} - \frac{9 y' \sin{\left(x^{2} \right)}}{2 \sqrt{y}} + \frac{15 e^{y^{3}}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 12 x^{\frac{3}{2}} y \cos{\left(x^{2} \right)} + 5 \sqrt{y} e^{y^{3}}}{3 \left(\sqrt{x} \sin{\left(x^{2} \right)} - 10 x y^{\frac{5}{2}} e^{y^{3}}\right)}