Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 x^{3} \sin{\left(y^{2} \right)} + 9 \log{\left(x \right)} \log{\left(y \right)}=38

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 4 x^{3} y y' \cos{\left(y^{2} \right)} - 6 x^{2} \sin{\left(y^{2} \right)} + \frac{9 y' \log{\left(x \right)}}{y} + \frac{9 \log{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 y \left(- 2 x^{3} \sin{\left(y^{2} \right)} + 3 \log{\left(y \right)}\right)}{x \left(4 x^{3} y^{2} \cos{\left(y^{2} \right)} - 9 \log{\left(x \right)}\right)}