Taking the Derivative of ln(x)^x: How-To & Steps

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we use a property of logarithms and their derivatives as well as the chain rule to find the derivative of the logarithm of x raised to the x power.

Finding the derivative of ln xx

We start by using the derivative of the natural logarithm of x.


We can look at this two ways:

  • the derivative of the natural logarithm of ''something'' is 1 over the ''something''
  • using the chain rule

The chain rule says the derivative of a function of a function is the derivative of the outer function multiplied by the derivative of the inner function. For ln x, the outer function is ln and its derivative is 1/x. The inner function is x and we write its derivative as d/dx of x. Thus, using the chain rule,


But the derivative of x with respect to x is 1:


And 1 times 1/x is simply 1/x:


This is the same result as just differentiating ln x, so why bother with the chain rule? Well, the task in this lesson has an inner function which is xx so using the chain rule on something simpler is a good warm up. Let's first work on the derivative of xx.

Step 1: Write y = xx.

It may seem arbitrary but let y = xx.

Step 2: Take the natural logarithm of both sides and simplify.

Taking the ln of both sides:


A property of logarithms allows us to bring the exponent out in front of the ln:


Step 3: Differentiate both sides.


The derivative of ln y with respect to x is 1/y times the derivative of y with respect to x. This is the left-hand side.

The right-hand side uses the product rule: the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

The derivative of the first function is the derivative of x which is 1.

The derivative of the second function is the derivative of ln x which is 1/x.


On the right-hand side we can further simplify because 1 times ln x is ln x and x times 1/x is 1:


Step 4: Solve for dy/dx

Multiplying both sides by y:


When we started, we let y = xx. On the right-hand side, replace y with xx:


On the left-hand side, replace y with xx:


And we have our derivative of the inner function.

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