Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle x \cos{\left(y \right)} - 27 \sqrt{y} \sin{\left(x^{3} \right)}=-25

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 81 x^{2} \sqrt{y} \cos{\left(x^{3} \right)} - x y' \sin{\left(y \right)} + \cos{\left(y \right)} - \frac{27 y' \sin{\left(x^{3} \right)}}{2 \sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(- 81 x^{2} y \cos{\left(x^{3} \right)} + \sqrt{y} \cos{\left(y \right)}\right)}{2 x \sqrt{y} \sin{\left(y \right)} + 27 \sin{\left(x^{3} \right)}}