Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 y^{2} \cos{\left(x \right)} + 9 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=-27

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 18 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 2 y^{2} \sin{\left(x \right)} + 18 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 4 y y' \cos{\left(x \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{9 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + y^{2} \sin{\left(x \right)}}{y \left(- 9 \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + 2 \cos{\left(x \right)}\right)}