Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 14 \sqrt{2} \sqrt{x} \sin{\left(y^{2} \right)} - y^{2} \sin{\left(x^{2} \right)}=-29

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 28 \sqrt{2} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 2 x y^{2} \cos{\left(x^{2} \right)} - 2 y y' \sin{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- x^{\frac{3}{2}} y^{2} \cos{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{2}}{y \left(\sqrt{x} \sin{\left(x^{2} \right)} - 14 \sqrt{2} x \cos{\left(y^{2} \right)}\right)}