Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 2 \sqrt{2} \sqrt{x} \sin{\left(y^{2} \right)} - 9 e^{x^{3}} \log{\left(y \right)}=-17$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 4 \sqrt{2} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 27 x^{2} e^{x^{3}} \log{\left(y \right)} - \frac{9 y' e^{x^{3}}}{y} + \frac{\sqrt{2} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{y \left(- 27 x^{\frac{5}{2}} e^{x^{3}} \log{\left(y \right)} + \sqrt{2} \sin{\left(y^{2} \right)}\right)}{9 \sqrt{x} e^{x^{3}} - 4 \sqrt{2} x y^{2} \cos{\left(y^{2} \right)}}$