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 3rd or 4th 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 kurtosis:

$Kurtosis(X) = E[(x - \mu_{X})^4] / \sigma^4$

again, not clear why raising $(x-\mu)^4$ gives "peakedness" or why $(x-\mu)^3$ should give skew. seems magical and mysterious.