Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 \log{\left(x \right)} \log{\left(y \right)} - 7 \sin{\left(x \right)} \sin{\left(y \right)}=47

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 7 y' \sin{\left(x \right)} \cos{\left(y \right)} - 7 \sin{\left(y \right)} \cos{\left(x \right)} + \frac{9 y' \log{\left(x \right)}}{y} + \frac{9 \log{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 7 x \sin{\left(y \right)} \cos{\left(x \right)} + 9 \log{\left(y \right)}\right)}{x \left(7 y \sin{\left(x \right)} \cos{\left(y \right)} - 9 \log{\left(x \right)}\right)}