Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 \sqrt{y} e^{x^{2}} - 9 \log{\left(y \right)} \sin{\left(x^{2} \right)}=34$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 16 x \sqrt{y} e^{x^{2}} - 18 x \log{\left(y \right)} \cos{\left(x^{2} \right)} - \frac{9 y' \sin{\left(x^{2} \right)}}{y} - \frac{4 y' e^{x^{2}}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{2 x \left(9 y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)} + 8 y^{2} e^{x^{2}}\right)}{9 \sqrt{y} \sin{\left(x^{2} \right)} + 4 y e^{x^{2}}}$