Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 12 \sqrt{2} \sqrt{x} \log{\left(y \right)} - 2 e^{x^{2}} e^{y^{2}}=41

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{12 \sqrt{2} \sqrt{x} y'}{y} - 4 x e^{x^{2}} e^{y^{2}} - 4 y y' e^{x^{2}} e^{y^{2}} + \frac{6 \sqrt{2} \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 2 x^{\frac{3}{2}} e^{x^{2} + y^{2}} + 3 \sqrt{2} \log{\left(y \right)}\right)}{2 \left(\sqrt{x} y^{2} e^{x^{2} + y^{2}} - 3 \sqrt{2} x\right)}