Finding the Derivative of xln(x)

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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will see how to use the product rule for derivatives to find the derivative of xln(x). Once we have found our derivative, we will look at how we can check our work and know we did everything correctly.

Steps to Solve

In order to find the derivative of xln(x), we are going to be using the product rule for derivatives. The product rule for derivatives states that to take the derivative of a product of functions, we multiply the derivative of the first function times the second function, and add it to the derivative of the second function multiplied by the first function. The following equation shows this in symbol form:


We can use this rule to find the derivative of xln(x) because this is a product of the functions f(x) = x and g(x) = ln(x). There are a couple more facts that we will need to know in order to find this derivative.

  • The derivative of x is 1. This comes from the fact that the derivative of xn is nxn-1. If we look at x as x1, we have that the derivative of x1 is 1 * x1 - 1 = x0 = 1.
  • The derivative of ln(x) is 1/x. This can be observed by looking at the slope of the tangent line at various values of x because the derivative of a function at a point is the slope of the tangent line at that point. Observe the following chart with the given slopes of the tangent line of ln(x) at various values of x:

x Slope of the tangent line of ln(x) at x
1 1
2 1/2
3 1/3
4 1/4
5 1/5

If we continue this pattern, we see that the slope of the tangent line of ln(x) at a given value of x is 1/x, so the derivative of ln(x) is 1/x.

These two facts, along with the product rule, will allow us to find the derivative of xln(x).

As we said, xln(x) is the product of f(x) = x and g(x) = ln(x). In order to use the product rule to find the derivative, we need to know f ' (x) and g ' (x).

From the facts, the derivative of x is 1, so f ' (x) = 1.

Also from the facts, the derivative of ln(x) is 1/x, so g ' (x) = 1/x.

Now we simply plug into the product rule for derivatives and simplify.


We see that the derivative of xln(x) is ln(x) + 1.

Checking Your Work with Integrals

After finding your derivative, you may want to have a way to know for sure that you found it correctly. Thankfully, we have a tool to check our work, and that tool is integrals! Integrals are a subject that are studied after derivatives are mastered. If you haven't studied them already, when you do you will learn that they have a close relationship with derivatives. In fact, integrals are called anti-derivatives, and they are related to derivatives in that if a is the derivative of b, then the integral of a is b + C, where C is a constant.


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