# Difference between revisions of "Second derivative"

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<math>\! f'' := (f')'</math> | <math>\! f'' := (f')'</math> | ||

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+ | ==Leibniz notation for second derivative== | ||

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+ | Suppose <math>f</math> is a function, and <math>x,y</math> are variables related by <math>y := f(x)</math>. Here, <math>x</math> is an ''independent variable'' and <math>y</math> is the ''dependent variable'' (with the dependency being described by the function <math>f</math>). We then define: | ||

+ | |||

+ | <math>\frac{d^2y}{dx^2} := f''(x)</math> | ||

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+ | This can also be written as: | ||

+ | |||

+ | <math>\frac{d^2}{dx^2}[f(x)]</math> | ||

+ | |||

+ | In particular, <math>d^2y/dx^2</math> is a ''function'' of <math>x</math>. Its value at <math>x = x_0</math> is defined as <math>f''(x_0)</math> and is denoted as follows: | ||

+ | |||

+ | <math>\! \frac{d^2y}{dx^2} |_{x = x_0} := f''(x_0)</math> |

## Revision as of 03:04, 18 December 2011

## 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:

## Leibniz notation for second derivative

Suppose is a function, and are variables related by . Here, is an *independent variable* and is the *dependent variable* (with the dependency being described by the function ). We then define:

This can also be written as:

In particular, is a *function* of . Its value at is defined as and is denoted as follows: