Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 3 \sqrt{y} \cos{\left(x^{3} \right)} + 4 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-17$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 9 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + 12 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + 12 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - \frac{3 y' \cos{\left(x^{3} \right)}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{x^{2} \left(8 \sqrt{y} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + 6 y \sin{\left(x^{3} \right)}\right)}{- 8 y^{\frac{5}{2}} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + \cos{\left(x^{3} \right)}}$