Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 \sin{\left(x \right)} \cos{\left(y \right)} - 9 \sin{\left(y \right)} \cos{\left(x \right)}=-25

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 2 y' \sin{\left(x \right)} \sin{\left(y \right)} - 9 y' \cos{\left(x \right)} \cos{\left(y \right)} + 9 \sin{\left(x \right)} \sin{\left(y \right)} - 2 \cos{\left(x \right)} \cos{\left(y \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{7 \cos{\left(x - y \right)} - 11 \cos{\left(x + y \right)}}{7 \cos{\left(x - y \right)} + 11 \cos{\left(x + y \right)}}