Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 e^{x^{2}} \log{\left(y \right)} - 9 \log{\left(x \right)} \sin{\left(y \right)}=-13

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 12 x e^{x^{2}} \log{\left(y \right)} - 9 y' \log{\left(x \right)} \cos{\left(y \right)} + \frac{6 y' e^{x^{2}}}{y} - \frac{9 \sin{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(4 x^{2} e^{x^{2}} \log{\left(y \right)} - 3 \sin{\left(y \right)}\right)}{x \left(3 y \log{\left(x \right)} \cos{\left(y \right)} - 2 e^{x^{2}}\right)}