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 \right)} + 7 x^{2} e^{y}=14

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 15 \sqrt{x} y' \sin{\left(y \right)} + 7 x^{2} y' e^{y} + 14 x e^{y} - \frac{15 \cos{\left(y \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 28 x^{\frac{3}{2}} e^{y} + 15 \cos{\left(y \right)}}{2 \left(7 x^{\frac{5}{2}} e^{y} + 15 x \sin{\left(y \right)}\right)}