Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 14 \sqrt{x} \cos{\left(y^{3} \right)} - 2 x^{2} \sqrt{y}=-23$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 42 \sqrt{x} y^{2} y' \sin{\left(y^{3} \right)} - \frac{x^{2} y'}{\sqrt{y}} - 4 x \sqrt{y} + \frac{7 \cos{\left(y^{3} \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- 4 x^{\frac{3}{2}} y + 7 \sqrt{y} \cos{\left(y^{3} \right)}}{x^{\frac{5}{2}} + 42 x y^{\frac{5}{2}} \sin{\left(y^{3} \right)}}$