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

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