Can someone provide an intuition on *why* the higher moments of a probability distribution $p_X$, like the third and fourth moments, correspond to skewness and kurtosis respectively?  Specifically, why does the deviation about the mean raised to the third or fourth power end up translating into a measure of skewness and kurtosis?  Is there a way to relate this to the third or fourth derivatives of the function?

Consider this definition of skewness and kurtosis:

$$\begin{matrix}
\text{Skewness}(X) = \mathbb{E}[(X - \mu_{X})^3] / \sigma^3, \\[6pt]
\text{Kurtosis}(X) = \mathbb{E}[(X - \mu_{X})^4] / \sigma^4. \\[6pt]
\end{matrix}$$

In these equations we raise the normalised value $(X-\mu)/\sigma$ to a power and take its expected value.  It is not clear to me why raising the normalised random variable to the power of four gives "peakedness" or why raising the normalised random variable to the power of three should give "skewness".  This seems magical and mysterious!