Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 \sqrt{x} \sqrt{y} + 8 y \cos{\left(x^{3} \right)}=-15

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{9 \sqrt{x} y'}{2 \sqrt{y}} - 24 x^{2} y \sin{\left(x^{3} \right)} + 8 y' \cos{\left(x^{3} \right)} + \frac{9 \sqrt{y}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 \left(16 x^{\frac{5}{2}} y^{\frac{3}{2}} \sin{\left(x^{3} \right)} - 3 y\right)}{16 \sqrt{x} \sqrt{y} \cos{\left(x^{3} \right)} + 9 x}