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

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