# Second derivative

## Contents

## Definition at a point

### Definition in terms of first derivative

The **second derivative** of a function at a point , denoted , is defined as the derivative at the point of the function defined as the derivative

Note that the *first* differentiation operation must be performed, not just at the point, but at all points near it, so that we have a *function* for the first derivative around the point, which we can then differentiate to calculate the second derivative *at* the point. It is *not* good enough to calculate the first derivative only *at* the particular point (i.e., to calculate *only* ) and then proceed to differentiate that; we need the value of the first derivative at nearby points too.

### Definition as a limit expression

The **second derivative** of a function at a point , denoted , is defined as follows:

More explicitly, this can be written as:

## Definition as a function

The **second derivative** of a function at a point is defined as the derivative of the derivative of the function. For a function , the second derivative is defined as: