Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 3 e^{y^{2}} \log{\left(x \right)} - 7 \cos{\left(x \right)} \cos{\left(y \right)}=44

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 6 y y' e^{y^{2}} \log{\left(x \right)} + 7 y' \sin{\left(y \right)} \cos{\left(x \right)} + 7 \sin{\left(x \right)} \cos{\left(y \right)} - \frac{3 e^{y^{2}}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{7 x \sin{\left(x \right)} \cos{\left(y \right)} - 3 e^{y^{2}}}{x \left(6 y e^{y^{2}} \log{\left(x \right)} - 7 \sin{\left(y \right)} \cos{\left(x \right)}\right)}