Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle e^{x} \log{\left(y \right)} - 8 \sin{\left(x \right)} \cos{\left(y^{2} \right)}=40

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 16 y y' \sin{\left(x \right)} \sin{\left(y^{2} \right)} + e^{x} \log{\left(y \right)} - 8 \cos{\left(x \right)} \cos{\left(y^{2} \right)} + \frac{y' e^{x}}{y} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- e^{x} \log{\left(y \right)} + 8 \cos{\left(x \right)} \cos{\left(y^{2} \right)}\right)}{16 y^{2} \sin{\left(x \right)} \sin{\left(y^{2} \right)} + e^{x}}