Finding Higher-Order Derivatives

What a Higher-Order Derivative Is

The first derivative tells you how a function is changing. But sometimes you need to know how that rate of change is changing. That's where higher-order derivatives come in.

Start with a function . Its first derivative is . If you differentiate the first derivative again, you get the second derivative, written as .

You can keep differentiating. The derivative of the second derivative is the third derivative, the derivative of the third derivative is the fourth derivative, and so on.

The notation may look intimidating, but the process is actually simple: differentiate, then differentiate the result again, and repeat as many times as necessary.

The Notation

There are several equivalent ways to write higher-order derivatives. The notation you should recognize on the AP exam includes:

For the second derivative, is especially common. For the third derivative, is common. You may also see prime notation such as .

A Step-by-Step Method

1. Find the first derivative. Apply the appropriate differentiation rules.
2. Differentiate the first derivative to obtain the second derivative.
3. Continue differentiating if a third, fourth, or higher derivative is requested.
4. Keep your notation organized. Label each derivative clearly so you don't accidentally differentiate the wrong expression.
5. Evaluate at a point only after finding the requested derivative, unless the problem specifically gives you another route.

Worked Examples

Higher-Order Derivatives in Motion

Higher-order derivatives have an important interpretation in AP Calculus. If represents the position of an object as a function of time, then the first derivative represents velocity.

The second derivative represents acceleration: it tells you how the velocity is changing.

The third derivative, sometimes called jerk, describes how acceleration is changing:

For AP Calculus, the most important connection is usually position → velocity → acceleration. If a problem gives you a position function and asks whether the object is speeding up or slowing down, the first and second derivatives are often the key.

Evaluating a Higher-Order Derivative at a Point

Sometimes the question doesn't ask you for the entire second derivative. Instead, it asks for its value at a particular point.

Common Mistakes

Recognizing Patterns

Some functions produce useful patterns when differentiated repeatedly. Recognizing these patterns can save time and help you check your work.

For sine and cosine, the derivatives cycle:

For an exponential function such as , every derivative is the same:

For a polynomial, repeated differentiation eventually produces zero. For example, a polynomial of degree has a zero derivative after enough differentiations.

Practice Problems