Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 6 x \cos{\left(y^{2} \right)} - 5 y e^{x}=-5$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 12 x y y' \sin{\left(y^{2} \right)} - 5 y e^{x} - 5 y' e^{x} + 6 \cos{\left(y^{2} \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- 5 y e^{x} + 6 \cos{\left(y^{2} \right)}}{12 x y \sin{\left(y^{2} \right)} + 5 e^{x}}$