Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 y^{2} e^{x} - 7 \log{\left(x \right)} \sin{\left(y \right)}=10

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 4 y^{2} e^{x} - 8 y y' e^{x} - 7 y' \log{\left(x \right)} \cos{\left(y \right)} - \frac{7 \sin{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{4 x y^{2} e^{x} + 7 \sin{\left(y \right)}}{x \left(8 y e^{x} + 7 \log{\left(x \right)} \cos{\left(y \right)}\right)}