Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle \sqrt{7} \sqrt{x} \cos{\left(y^{2} \right)} - 7 y^{3} \cos{\left(x^{2} \right)}=-19

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 2 \sqrt{7} \sqrt{x} y y' \sin{\left(y^{2} \right)} + 14 x y^{3} \sin{\left(x^{2} \right)} - 21 y^{2} y' \cos{\left(x^{2} \right)} + \frac{\sqrt{7} \cos{\left(y^{2} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{28 x^{\frac{3}{2}} y^{3} \sin{\left(x^{2} \right)} + \sqrt{7} \cos{\left(y^{2} \right)}}{2 y \left(21 \sqrt{x} y \cos{\left(x^{2} \right)} + 2 \sqrt{7} x \sin{\left(y^{2} \right)}\right)}