Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 12 \sqrt{2} \sqrt{x} \log{\left(y \right)} + 4 \sin{\left(x^{2} \right)} \cos{\left(y^{3} \right)}=21

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{12 \sqrt{2} \sqrt{x} y'}{y} + 8 x \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - 12 y^{2} y' \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} - \frac{6 \sqrt{2} \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(4 x^{\frac{3}{2}} \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - 3 \sqrt{2} \log{\left(y \right)}\right)}{6 \left(\sqrt{x} y^{3} \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + \sqrt{2} x\right)}