Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 3 x \log{\left(y \right)} + 8 y \cos{\left(x^{2} \right)}=31$

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