Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 24 \sqrt{x} \sin{\left(y^{3} \right)} + 6 \sin{\left(x^{3} \right)} \cos{\left(y^{2} \right)}=-4

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 72 \sqrt{x} y^{2} y' \cos{\left(y^{3} \right)} + 18 x^{2} \cos{\left(x^{3} \right)} \cos{\left(y^{2} \right)} - 12 y y' \sin{\left(x^{3} \right)} \sin{\left(y^{2} \right)} + \frac{12 \sin{\left(y^{3} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{\frac{3 x^{\frac{5}{2}} \cos{\left(x^{3} \right)} \cos{\left(y^{2} \right)}}{2} + \sin{\left(y^{3} \right)}}{y \left(\sqrt{x} \sin{\left(x^{3} \right)} \sin{\left(y^{2} \right)} - 6 x y \cos{\left(y^{3} \right)}\right)}