Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 e^{y^{2}} \cos{\left(x \right)} - 7 \log{\left(x \right)} \cos{\left(y \right)}=-47

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