Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 2 \sqrt{2} \sqrt{y} \sin{\left(x^{2} \right)} - \log{\left(y \right)} \cos{\left(x^{2} \right)}=-1

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