Higher-Order Derivatives

Meaning of Higher-Order Derivatives

Calculus is all about change. When finding the derivative of a function, you are left with yet another function, which in turn, can be differentiated again under certain circumstances.

The derivatives of functions that have already been differentiated are known as Higher-Order Derivatives.

Take for instance, the derivative of the polynomial function

\[ f(x) = x^5 +2x^4 -x^2 +4x +1,\]

for which you can use the Power Rule to find its derivative, that is

\[ f'(x) = 5x^4 +8x^3 -2x +4.\]

The derivative you just found is another polynomial function, so you can use The Power Rule again to find the derivative of \( f'(x), \) obtaining

\[(f'(x))' = 20x^3+24x^2-2.\]

This function you just found is known as the second derivative of \( f.\) If you keep this notation, you can notice that there are a lot of parentheses. Imagine what will happen if you find the third or the fourth derivative! To address this issue, there are two different notations that can be used when writing Higher-Order Derivatives.

Rules for Writing Higher-Order Derivatives

More than rules for writing Higher-Order Derivatives, we talk about notation. Having consistent notation is useful so everyone is on the same page when talking about Higher-Order Derivatives.

Prime Notation for Higher-Order Derivatives

Consider the previous example of a second derivative,

\[ \left( f'(x)\right)' = 20x^3 +24x^2 -2.\]

Rather than using all these parentheses, the primes are placed next to each other, that is

\[ f''(x) = 20x^3 +24x^2 -2.\]

If you differentiate the function once again, you will have a third derivative. This is denoted by using three primes

\[f'''(x) = 60x^2 +48x.\]

Before proceeding to the fourth derivative, note that if you keep adding primes this would be extremely cumbersome. Rather than adding more primes, a number in placed between parentheses.

\[f^{(4)}(x) = 120x +48.\]

This notation is kept for further derivatives.

Leibniz's Notation for Higher-Order Derivatives

Leibniz's notation uses fractional notation rather than primes.

Once again, consider the function of the previous example. Its derivative using Leibniz's notation is written as

\[ \frac{\mathrm{d}f}{\mathrm{d}x} = 5x^4 +8x^3 -2x +4. \]

For the second derivative, you need to place the number 2 in the following way:

\[ \frac{\mathrm{d}^2 f}{\mathrm{d}x^2} = 20x^3 +24x^2 -2. \]

This notation is the same for all the higher-order derivatives, so

\[ \frac{\mathrm{d}^3f}{\mathrm{d}x^3} = 60x^2 +48x,\]

and

\[ \frac{\mathrm{d}^4f}{\mathrm{d}x^4} = 120x +48\]

You can read the notation as follows: The fourth derivative of \( f \) with respect to \( x \) four times.

\[\frac{\mathrm{d}^4 f}{\mathrm{d}x^4}\]

This makes more sense when studying Partial Derivatives, but this is out of the scope of this article!

Formulas for Higher-Order Derivatives

Most of the time, there is not a formula for finding Higher-Order Derivatives. However, some functions have patterns that can be followed when differentiated.

Consider the following exponential function

\[f(x)=e^{2x}.\]

You can find its derivative by using the Chain Rule, which gives you

\[f'(x)=2e^{2x.}\]

For its second derivative you can use the Constant Multiple Rule to obtain

\[ \begin{align} f''(x) &= 2\left( 2e^{2x} \right) \\ &= 2^{2} e^{2x} \\ &= 4e^{2x}.\end{align}\]

Each further differentiation will multiply the previous derivative by a factor of two, so its fifth derivative for example would be

\[ \begin{align} f^{(5)}(x) &= 2^{5} e^{2x} \\ &= 32e^{2x}. \end{align} \]

In general, its \(n\)^th derivative is given by

\[ f^{(n)} (x) = 2^n e^{2x}.\]

There are more functions whose derivatives relate perfectly to certain patterns, but this is also out of the scope of this article.

Examples of Higher-Order Derivatives

Finding higher-order derivatives is a straightforward task. Just keep differentiating the function as many times as you need! Here are some examples.

You used The Power Rule three times in a row in the above example! You will not always need to use the same differentiation rule over and over again, here is another example.

Let's take a look at the derivative of another polynomial function.

Applications of higher-order derivatives

Applications of Higher-Order Derivatives in Mathematics

The second derivative of a function gives us information about the maxima and minima of a function and its concavity. For further information about these topics, you can check out our articles on the following subjects:

• Maxima and Minima

• The Second Derivative Test

• Derivatives and the Shape of a Graph

Applications of Higher-Order Derivatives in Science

A prime example of higher-order derivatives is acceleration.

This means that if you have a function that describes the position of an object, then its second derivative will describe its acceleration.

Second-order derivatives are also present in more natural phenomena. These include, but are not limited to:

• The diffusion of heat through a material.
• The propagation of waves.
• The diffusion of a substance in a liquid.
• The mechanics of fluids.