Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 \sqrt{2} \sqrt{y} \log{\left(x \right)} - 7 y \cos{\left(x^{3} \right)}=2

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 21 x^{2} y \sin{\left(x^{3} \right)} - 7 y' \cos{\left(x^{3} \right)} - \frac{4 \sqrt{2} y' \log{\left(x \right)}}{\sqrt{y}} - \frac{8 \sqrt{2} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{21 x^{3} y^{\frac{3}{2}} \sin{\left(x^{3} \right)} - 8 \sqrt{2} y}{x \left(7 \sqrt{y} \cos{\left(x^{3} \right)} + 4 \sqrt{2} \log{\left(x \right)}\right)}