SL 5.7—The second derivative

When you differentiate a function $f(x)$, you get $f^{\prime }(x)$ — the rate of change of $f$. But what if you differentiate again? The result is the second derivative, which measures how the rate of change is itself changing.

The two standard notations you need to know are:

• Lagrange notation: $f^{\prime \prime }(x)$
• Leibniz notation: $\frac{d^{2}y}{d{x}^2}$

Think of it this way: if $f(x)$ is position, then $f^{\prime }(x)$ is velocity, and $f^{\prime \prime }(x)$ is acceleration. The second derivative is asking "how fast is the slope changing?"