Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 10 \sqrt{2} \sqrt{x} \sin{\left(y^{2} \right)} - 9 y^{3} \cos{\left(x \right)}=-24

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 20 \sqrt{2} \sqrt{x} y y' \cos{\left(y^{2} \right)} + 9 y^{3} \sin{\left(x \right)} - 27 y^{2} y' \cos{\left(x \right)} - \frac{5 \sqrt{2} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{9 \sqrt{x} y^{3} \sin{\left(x \right)} - 5 \sqrt{2} \sin{\left(y^{2} \right)}}{y \left(27 \sqrt{x} y \cos{\left(x \right)} + 20 \sqrt{2} x \cos{\left(y^{2} \right)}\right)}