It's well-known that $f'(x),f''(x)$ determine the monotonicity and the concavity of $f(x)$ respectively. Besides, we can see $f'(x)$ and $f''(x)$ from the graph of $f(x)$.
But how about $f'''(x)$ and other higher orderd derivative? What's its geometrical meaning on the graph of $f(x)$? Or, limited by our human-being's perception, we can't see high-orderd derivative?
Usually we don't give a specific geometrical meaning to $f'''(x)$ with reference to $f(x)$ but we can observe that $$f'''(x)=(f''(x))'=(f'(x))''$$
and therefore $f'''(x)$ represents
With reference to $f(x)$, we can be interested in the higher order derivatives to determine the nature of critical points.
Notably, when derivatives exist if $f'(x_0)=0$ and $\exists k \geq 2$ s.t. $f^k(x_0) \neq 0$ then
when k is even we have a max/min in $x_0$ (depending on the sign)
when k is odd we have an inflection point in $x_0$
If $f(t)$ represents position at time $t$, then the third derivative $f'''(t)$ is sometimes called jerk and represents change in acceleration at time $t$. See this link on Wikipedia for more information.
