Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 \sqrt{3} \sqrt{y} \log{\left(x \right)} - 5 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=4

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 10 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} - 10 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - \frac{2 \sqrt{3} y' \log{\left(x \right)}}{\sqrt{y}} - \frac{4 \sqrt{3} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{5 x^{2} \sqrt{y} \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 2 \sqrt{3} y}{x \left(5 y^{\frac{3}{2}} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + \sqrt{3} \log{\left(x \right)}\right)}