Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 10 \sqrt{2} \sqrt{y} \cos{\left(x \right)} - 4 e^{y^{2}} \log{\left(x \right)}=43

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 10 \sqrt{2} \sqrt{y} \sin{\left(x \right)} - 8 y y' e^{y^{2}} \log{\left(x \right)} + \frac{5 \sqrt{2} y' \cos{\left(x \right)}}{\sqrt{y}} - \frac{4 e^{y^{2}}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(5 \sqrt{2} x y \sin{\left(x \right)} + 2 \sqrt{y} e^{y^{2}}\right)}{x \left(- 8 y^{\frac{3}{2}} e^{y^{2}} \log{\left(x \right)} + 5 \sqrt{2} \cos{\left(x \right)}\right)}