Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 2 y^{2} \log{\left(x \right)} - 2 \sin{\left(x^{3} \right)} \cos{\left(y^{2} \right)}=-5

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