Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - \sqrt{5} \sqrt{x} y^{3} + 2 \log{\left(y \right)} \cos{\left(x \right)}=-10

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