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Finding the Second Derivative: Formula & Examples

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you will learn the two-step process involved in finding the second derivative. Also, look at some examples to get your feet wet before jumping into the quiz.

Formula

The second derivative is defined as the derivative of the first derivative. Its symbol is the function followed by two apostrophe marks.

The second derivative symbol.
second derivative

The formula for calculating the second derivative is this.

The second derivative formulal.
second derivative

What this formula tells you to do is to first take the first derivative. Then you would take the derivative of the first derivative to find your second derivative. I have omitted the (x) next to the f as that would have made the notation more difficult to read.

Let's work it out with an example to see it in action. Let's find the second derivative of the following function.

Find the second derivative of this function.
second derivative

Step One: Find the First Derivative

Our first step is to take the first derivative of our function. Our function is a polynomial, so we will calculate the derivative of each term by using the power rule. As you recall, a polynomial is simply a mathematical expression that contains both constants and variables.

Taking the first derivative.
second derivative

Recall that for each term, the power rule tells you to multiply the number by the exponent and then you reduce the exponent by 1. So for our first term, x^4, we multiply our number 1 by the 4 to get 4. We then reduce the exponent by 1 to get 3. Thus, our first derivative for our first term is 4x^3. We would do likewise to our second term, 3x^3. We multiply the number 3 by the exponent 3 to get 9. We then reduce the exponent by 1 to get 2. Thus, our first derivative of the second term is 9x^2. Doing likewise to the third term, we get 16x.

Now that we have gotten our first derivative, let's continue on to our next step.

Step Two: Find the Second Derivative

Our first derivative is still a polynomial, so we will continue to use the power rule for each term to take the derivative of this function.

Taking the second derivative.
second derivative

Once we have taken the derivative of the first derivative, we can stop. This is our answer. Following the power rule, we see that our first term of the first derivative, 4x^3, becomes 12x^2 since we multiply the 4 and the 3 and drop the exponent by 1. The second term, 9x^2, we have done the same by multiplying the 9 and the 2 and dropping the exponent by 1 also. The last term, 16x, has an exponent of 1, so we multiply the 16 by 1 and then we drop the exponent from 1 to 0, turning the x into a 1. Our last term for our second derivative is then 16.

Let's try a few more examples.

Examples

Let's try taking the second derivative of this function.

Find the second derivative.
second derivative

This is a simple function. So, tell me what should we do first? Yes, we should take the first derivative of this function. Which rule do we need to apply? Yes, we need to apply the power rule again. Let's go ahead and do that to see what we get.

The first derivative.
second derivative

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