Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - \sqrt{5} \sqrt{x} \sin{\left(y^{3} \right)} - 9 \log{\left(y \right)} \cos{\left(x^{2} \right)}=-40

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 3 \sqrt{5} \sqrt{x} y^{2} y' \cos{\left(y^{3} \right)} + 18 x \log{\left(y \right)} \sin{\left(x^{2} \right)} - \frac{9 y' \cos{\left(x^{2} \right)}}{y} - \frac{\sqrt{5} \sin{\left(y^{3} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(36 x^{\frac{3}{2}} \log{\left(y \right)} \sin{\left(x^{2} \right)} - \sqrt{5} \sin{\left(y^{3} \right)}\right)}{6 \left(3 \sqrt{x} \cos{\left(x^{2} \right)} + \sqrt{5} x y^{3} \cos{\left(y^{3} \right)}\right)}