Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 \sqrt{7} \sqrt{x} \sin{\left(y^{2} \right)} + 5 \log{\left(x \right)} \log{\left(y \right)}=20

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 8 \sqrt{7} \sqrt{x} y y' \cos{\left(y^{2} \right)} + \frac{5 y' \log{\left(x \right)}}{y} + \frac{5 \log{\left(y \right)}}{x} - \frac{2 \sqrt{7} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 5 \sqrt{x} \log{\left(y \right)} + 2 \sqrt{7} x \sin{\left(y^{2} \right)}\right)}{5 x^{\frac{3}{2}} \log{\left(x \right)} - 8 \sqrt{7} x^{2} y^{2} \cos{\left(y^{2} \right)}}