Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 4 e^{x^{3}} \cos{\left(y \right)} + 2 \log{\left(y \right)} \sin{\left(x^{2} \right)}=-42$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 12 x^{2} e^{x^{3}} \cos{\left(y \right)} + 4 x \log{\left(y \right)} \cos{\left(x^{2} \right)} - 4 y' e^{x^{3}} \sin{\left(y \right)} + \frac{2 y' \sin{\left(x^{2} \right)}}{y} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 x y \left(3 x e^{x^{3}} \cos{\left(y \right)} + \log{\left(y \right)} \cos{\left(x^{2} \right)}\right)}{2 y e^{x^{3}} \sin{\left(y \right)} - \sin{\left(x^{2} \right)}}$