Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 7 x^{2} \log{\left(y \right)} + 3 e^{y} \sin{\left(x \right)}=8$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{7 x^{2} y'}{y} - 14 x \log{\left(y \right)} + 3 y' e^{y} \sin{\left(x \right)} + 3 e^{y} \cos{\left(x \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{y \left(- 14 x \log{\left(y \right)} + 3 e^{y} \cos{\left(x \right)}\right)}{7 x^{2} - 3 y e^{y} \sin{\left(x \right)}}$