Even-Order and Odd-Order Derivatives

Even-order and odd-order derivatives are successive derivatives of a function, classified according to how many times the function has been differentiated. Even-order derivatives have orders 2, 4, 6, and so on, while odd-order derivatives have orders 1, 3, 5, and so on.

A function can be differentiated more than once, provided each successive derivative exists. This allows us to study how a function behaves beyond its initial rate of change.

The first derivative is obtained by differentiating the original function. Differentiating the first derivative gives the second derivative. Differentiating the second gives the third, and the process continues in the same way.

For a function \(f(x)\), the first four derivatives are written as follows:

• First derivative: \(f'(x)\), odd order.
• Second derivative: \(f''(x)\), even order.
• Third derivative: \(f'''(x)\), odd order.
• Fourth derivative: \(f^{(4)}(x)\), even order.

The important point is that the order of a derivative refers to the number of differentiations performed. It does not depend on whether the derivative is positive, negative, or zero.

Example of Even-Order and Odd-Order Derivatives

Let's see how this works with a polynomial function.

Consider

\[ f(x)=x^4 \]

We can find its successive derivatives by differentiating one step at a time.

The first derivative is

\[ f'(x)=4x^3 \]

This is an odd-order derivative because the original function has been differentiated once.

Differentiating again gives the second derivative.

\[ f''(x)=12x^2 \]

This is an even-order derivative because we have differentiated the original function twice.

Next, we calculate the third derivative.

\[ f'''(x)=24x \]

Finally, the fourth derivative is

\[ f^{(4)}(x)=24 \]

We can now identify the pattern. The first and third derivatives are of odd order, while the second and fourth derivatives are of even order.

Note. In this example, the fourth derivative is a constant. Differentiating it again gives zero. Therefore, all derivatives of order greater than four are identically zero.

Why Are Even-Order and Odd-Order Derivatives Useful?

The distinction between even-order and odd-order derivatives becomes especially useful when studying the stationary points of a function.

A stationary point occurs where the first derivative is zero. At such a point, the graph has a horizontal tangent, provided the function is differentiable there.

However, a horizontal tangent does not necessarily mean that the function has reached a maximum or minimum.

A stationary point may correspond to:

• A local maximum, where the function's value is greater than or equal to nearby values.
• A local minimum, where the function's value is less than or equal to nearby values.
• A point that is neither a local maximum nor a local minimum, such as a stationary point of inflection.

In many cases, the second derivative test can tell us whether a stationary point is a maximum or minimum. But what happens when the second derivative is also zero?

That's when higher-order derivatives become useful.

How to Identify a Local Maximum, Local Minimum, or Inflection Point

Suppose the first derivative vanishes at \(x_0\).

\[ f'(x_0)=0 \]

We know that \(x_0\) is a stationary point, but we still need to determine its nature.

To do this, we examine the successive derivatives.

• If the second derivative is nonzero at \(x_0\), its sign determines whether the function has a local maximum or a local minimum.
• If the second derivative is also zero at \(x_0\), we calculate the third derivative. If that is zero as well, we continue with higher-order derivatives until we find the first one that is nonzero.

Provided the function is sufficiently differentiable in a neighborhood of \(x_0\), the order and sign of the first nonzero derivative allow us to classify the stationary point.

This procedure is called the higher-order derivative test, also known as the nth derivative test.

There are two main cases.

1] The First Nonzero Derivative Is of Even Order

Suppose the first nonzero derivative is of even order, such as the second, fourth, or sixth derivative.

In this case, the function has a local extremum at \(x_0\). This means that the point corresponds to either a local maximum or a local minimum.

To determine which one, we examine the sign of the first nonzero derivative.

• If the derivative is positive, the function has a local minimum at \(x_0\).
• If the derivative is negative, the function has a local maximum at \(x_0\).

In other words, an even-order first nonzero derivative indicates a local extremum, while its sign tells us whether that extremum is a maximum or minimum.

2] The First Nonzero Derivative Is of Odd Order

Now suppose the first nonzero derivative is of odd order greater than one, such as the third or fifth derivative.

In this case, the function has neither a local maximum nor a local minimum at \(x_0\).

Instead, the graph has a stationary point of inflection, also called a horizontal inflection point.

At this point, the tangent line is horizontal, but the function continues through the level \(f(x_0)\) rather than turning back. The graph also changes concavity as it passes through the point.

This distinction is important because a stationary point does not always correspond to an extremum.

Worked Examples

Let's apply the higher-order derivative test to three functions. These examples show how the order and sign of the first nonzero derivative determine the behavior of a function near a stationary point.

Example 1. A Local Minimum

Consider the function

\[ f(x)=x^4 \]

We want to determine the nature of the stationary point at \(x=0\).

First, calculate the derivatives.

\[ f'(x)=4x^3 \]

\[ f''(x)=12x^2 \]

\[ f'''(x)=24x \]

\[ f^{(4)}(x)=24 \]

At \(x=0\), the first three derivatives are zero.

\[ f'(0)=f''(0)=f'''(0)=0 \]

The fourth derivative, however, is positive.

\[ f^{(4)}(0)=24>0 \]

The first nonzero derivative is therefore the fourth derivative, which is of even order.

Since this derivative is positive, the function has a local minimum at \(x=0\).

In fact, \(f(x)=x^4\) has its absolute minimum at the origin, the point \((0,0)\) on the graph.

Example 2. A Local Maximum

Now consider the function

\[ f(x)=-x^4 \]

Let's calculate its successive derivatives.

\[ f'(x)=-4x^3 \]

\[ f''(x)=-12x^2 \]

\[ f'''(x)=-24x \]

\[ f^{(4)}(x)=-24 \]

As in the previous example, the first three derivatives vanish at \(x=0\).

\[ f'(0)=f''(0)=f'''(0)=0 \]

This time, however, the fourth derivative is negative.

\[ f^{(4)}(0)=-24<0 \]

The first nonzero derivative is again of even order, but its sign is negative.

Therefore, the function has a local maximum at \(x=0\).

In fact, \(f(x)=-x^4\) has its absolute maximum at the origin, the point \((0,0)\) on the graph.

Notice the difference between the first two examples. Both functions have a fourth-order first nonzero derivative, but the opposite signs lead to opposite types of extrema.

Example 3. A Stationary Point of Inflection

Finally, consider the function

\[ f(x)=x^3 \]

Let's calculate its first three derivatives.

\[ f'(x)=3x^2 \]

\[ f''(x)=6x \]

\[ f'''(x)=6 \]

At \(x=0\), both the first and second derivatives vanish.

\[ f'(0)=f''(0)=0 \]

The third derivative, however, is nonzero.

\[ f'''(0)=6 \]

The first nonzero derivative is the third derivative, which is of odd order.

Therefore, the function has neither a local maximum nor a local minimum at \(x=0\).

Instead, the graph has a stationary point of inflection.

The graph of \(f(x)=x^3\) passes through the origin with a horizontal tangent. As it crosses the origin, its concavity changes.

This example shows why a zero first derivative does not necessarily indicate a maximum or minimum.

Why Do Even-Order and Odd-Order Derivatives Lead to Different Results?

The reason lies in a fundamental property of even and odd powers.

An even power of a real number is always nonnegative. For example, consider

\[ f(x)=x^4 \]

Whether \(x\) is positive or negative, \(x^4\) is positive whenever \(x\ne 0\). At the origin, its value is zero.

An odd power behaves differently. Consider

\[ f(x)=x^3 \]

In this case, the sign depends on the sign of \(x\).

• When \(x<0\), the value of \(x^3\) is negative.
• When \(x>0\), the value of \(x^3\) is positive.

This difference helps explain the behavior of functions near their stationary points.

More precisely, suppose the lower-order derivatives vanish at \(x_0\), and the first nonzero derivative is of order \(n\).

If the function is sufficiently differentiable, Taylor's theorem gives the following local expansion:

\[ f(x)-f(x_0)=\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o\left((x-x_0)^n\right) \]

The first term on the right-hand side is the leading nonconstant term. It determines the sign of \(f(x)-f(x_0)\) for values of \(x\) sufficiently close to \(x_0\), with \(x\ne x_0\).

If \(n\) is even, the power \((x-x_0)^n\) is positive on both sides of \(x_0\). As a result, the leading term has the same sign on either side of the stationary point.

Its sign depends on \(f^{(n)}(x_0)\).

• If \(f^{(n)}(x_0)>0\), the nearby values of the function are greater than \(f(x_0)\), so the function has a local minimum.
• If \(f^{(n)}(x_0)<0\), the nearby values are less than \(f(x_0)\), so the function has a local maximum.

If \(n\) is odd, the power \((x-x_0)^n\) changes sign as \(x\) passes through \(x_0\).

Consequently, the function takes values on opposite sides of \(f(x_0)\) in every sufficiently small neighborhood of the point. This rules out a local maximum or minimum.

When \(n\) is odd and greater than one, the graph also changes concavity at \(x_0\), giving a stationary point of inflection.

This is the mathematical basis of the higher-order derivative test. The parity of the first nonzero derivative determines whether the stationary point is an extremum or an inflection point, while the derivative's sign provides further information about the function's local behavior.

Even-Order Derivatives vs. Even Functions

One final distinction is worth keeping in mind: an even-order derivative is not the same thing as an even function.

Although the terms sound similar, they describe two different mathematical properties.

A derivative is of even order when the original function has been differentiated an even number of times.

An even function, by contrast, is defined by a symmetry property. A function \(f(x)\) is even if

\[ f(-x)=f(x) \]

for every \(x\) in its domain, which must be symmetric about the origin.

Geometrically, this means that the graph of an even function is symmetric about the y-axis.

Example

Consider the function

\[ f(x)=x^3+x \]

This function is odd because

\[ f(-x)=-f(x) \]

Now calculate its derivatives.

The first derivative is

\[ f'(x)=3x^2+1 \]

The second derivative is

\[ f''(x)=6x \]

The second derivative is an even-order derivative because the original function has been differentiated twice.

However, the resulting function \(f''(x)=6x\) is odd, since

\[ f''(-x)=-f''(x) \]

So an even-order derivative does not have to be an even function.

The key distinction is that derivative order describes how many times a function has been differentiated, while function parity describes symmetry with respect to the y-axis or the origin.

These concepts are independent, and the same distinction applies to derivatives of any order.