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

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 10 x e^{x^{2}} \cos{\left(y^{2} \right)} - 10 y y' e^{x^{2}} \sin{\left(y^{2} \right)} - \frac{7 y' \log{\left(x \right)}}{y} - \frac{7 \log{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(10 x^{2} e^{x^{2}} \cos{\left(y^{2} \right)} - 7 \log{\left(y \right)}\right)}{x \left(10 y^{2} e^{x^{2}} \sin{\left(y^{2} \right)} + 7 \log{\left(x \right)}\right)}