I like a different geometric interpretation from the graphical one. If you think about a function f:R-->R as a transformation on the real line R, then the interpretation of f'(x) is that it's the scaling factor of this transformation near the point x.

This interpretation is good for a geometric understanding of the change-of-variables formula for integrals. It also makes the chain rule pretty plain.

I like a different geometric interpretation from the graphical one. If you think about a function $f : \mathbb{R} \to \mathbb{R}$ as a transformation on the real line $\mathbb{R}$, then the interpretation of $f'(x)$ is that it's the scaling factor of this transformation near the point $x$.

This interpretation is good for a geometric understanding of the change-of-variables formula for integrals. It also makes the chain rule pretty plain.