Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 7 e^{x^{2}} \sin{\left(y \right)} + 6 \log{\left(x \right)} \cos{\left(y^{2} \right)}=38

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 14 x e^{x^{2}} \sin{\left(y \right)} - 12 y y' \log{\left(x \right)} \sin{\left(y^{2} \right)} - 7 y' e^{x^{2}} \cos{\left(y \right)} + \frac{6 \cos{\left(y^{2} \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(- 7 x^{2} e^{x^{2}} \sin{\left(y \right)} + 3 \cos{\left(y^{2} \right)}\right)}{x \left(12 y \log{\left(x \right)} \sin{\left(y^{2} \right)} + 7 e^{x^{2}} \cos{\left(y \right)}\right)}