Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 7 \sqrt{2} \sqrt{x} \cos{\left(y^{2} \right)} - 2 y^{3} \cos{\left(x^{2} \right)}=-48

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 14 \sqrt{2} \sqrt{x} y y' \sin{\left(y^{2} \right)} + 4 x y^{3} \sin{\left(x^{2} \right)} - 6 y^{2} y' \cos{\left(x^{2} \right)} - \frac{7 \sqrt{2} \cos{\left(y^{2} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{8 x^{\frac{3}{2}} y^{3} \sin{\left(x^{2} \right)} - 7 \sqrt{2} \cos{\left(y^{2} \right)}}{4 y \left(3 \sqrt{x} y \cos{\left(x^{2} \right)} - 7 \sqrt{2} x \sin{\left(y^{2} \right)}\right)}