# What does second-order derivative really tell you geometrically?

I understand the first-order derivative (or partial derivative) tells you the slope of the tangent line at that point. If this is the case, what does second-order derivative tell you about?

Thank you

• concavity: if it is concave up or concave down (you are familiar with this through open upwards and downwards parabolas) – user29418 Feb 13 '18 at 21:31
• Right. I just remember now. Thank you – Katherine Feb 13 '18 at 21:32

• $f''(x)\ge 0\implies f(x)$ is convex
• $f''(x)\le 0 \implies f(x)$ is concave