Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle \sqrt{5} \sqrt{y} \cos{\left(x^{2} \right)} - 8 y \sin{\left(x \right)}=-21

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 2 \sqrt{5} x \sqrt{y} \sin{\left(x^{2} \right)} - 8 y \cos{\left(x \right)} - 8 y' \sin{\left(x \right)} + \frac{\sqrt{5} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{4 \left(\sqrt{5} x y \sin{\left(x^{2} \right)} + 4 y^{\frac{3}{2}} \cos{\left(x \right)}\right)}{- 16 \sqrt{y} \sin{\left(x \right)} + \sqrt{5} \cos{\left(x^{2} \right)}}