Let $f(x) = 1 + x + x^{2} + x^{3}$. Then the derivative is Thus, $f$ is monotone on $\mathbb{R}$ because

Hence $f$ has an inverse function. Since $f(x)\to\infty$ as $x\to\infty$, and $f(x)\to-\infty$ as $x\to-\infty$, it follows that the inverse function $f^{-1}$ is defined for all numbers.

Find the derivative:

$D_x \left( f^{-1} \right)(40) =$