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

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