Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 9 \sqrt{x} y^{3} + 4 \sqrt{y} \sin{\left(x^{3} \right)}=21$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 27 \sqrt{x} y^{2} y' + 12 x^{2} \sqrt{y} \cos{\left(x^{3} \right)} + \frac{2 y' \sin{\left(x^{3} \right)}}{\sqrt{y}} + \frac{9 y^{3}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{24 x^{\frac{5}{2}} y \cos{\left(x^{3} \right)} + 9 y^{\frac{7}{2}}}{4 \sqrt{x} \sin{\left(x^{3} \right)} + 54 x y^{\frac{5}{2}}}$