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.

Second Derivative: The second derivative of a function is obtained by differentiating the function twice. It describes how the first derivative (the rate of change) is changing. It is denoted $f^{\prime \prime }(x)$ in Lagrange notation, or $\frac{d^{2}y}{d{x}^2}$ in Leibniz notation.

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?"