Logarithmic Differentiation

What’s the derivative of ? It looks like it should be easy, but none of the rules so far fit: it isn’t a power of with a fixed exponent, and it isn’t a fixed number raised to the . Logarithmic differentiation handles it in three lines, and the same trick turns big products and quotients, which would take a page of product and quotient rules, into a quick sum.

Why the Usual Rules Don’t Work on xˣ

You already have a rule for a variable in the base, the power rule: . And you have one for a variable in the exponent, from the derivatives of exponential functions: . But each rule depends on the other part being a constant. In both parts change, so neither rule applies.

The way out is a logarithm. Taking brings the exponent down to the ground floor, where it’s just a factor you can handle with the product rule.

How to Do Logarithmic Differentiation: 4 Steps

1. Take of both sides of .
2. Expand with the logarithm rules: , , and .
3. Differentiate both sides with respect to . The left side, , becomes by the chain rule, just as in implicit differentiation.
4. Multiply both sides by , then replace with the original function.

Step 4 is the one people forget. After step 3 you’ve found , not .

Example 1: The Derivative of x to the x

Find for , where .

Take logs and bring the exponent down:

Differentiate. The right side is a product, so use the product rule:

Multiply by :

The graph confirms it. At the formula gives , and the tangent line there does have slope 1. The derivative is zero when , that is at , and that’s exactly where the curve bottoms out.

Here’s a nice check on the answer. Expand it: . The first piece is what the exponential rule would have given, and the second is what the power rule would have given. Each wrong method gets half the answer, and the true derivative is the two halves added together.

Example 2: A Messy Product and Quotient

Find for .

The quotient rule would work, with the product rule inside the numerator and the chain rule inside that. Logs are faster. Expand completely before differentiating:

Each term is now a simple log, so differentiate term by term:

Multiply by :

Leaving the answer in this ” times a bracket” form is standard. Every factor of the original function shows up as one simple term in the bracket.

Example 3: A Variable Power of sin x

Find for , where .

Product rule on the right:

Example 4: A Function Raised to a Power of x

Find for .

The product rule, with the chain rule for :

Example 5: Where Is x^(1/x) Largest?

Find where has a horizontal tangent line.

This is the th root of : , , . It seems to peak somewhere around 3. Logarithmic differentiation finds exactly where:

So . Since is never zero, only when , at . The peak value is . The number turns up again.

Example 6: The Power Rule for Any Exponent

The power rule lesson said that holds for every real , even irrational ones like , but left the proof for later. Here it is, for . Let :

Three lines, and it works for any constant at all.

The Logarithmic Derivative

The quantity that appears in step 3 has a name: the logarithmic derivative of , because it equals . It measures the relative rate of change, how fast changes as a fraction of its current size. That’s why it shows up in growth rates: a population with is growing by 3 percent per unit of time.

Its most useful property is the one Example 2 relied on: the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. For ,

Multiply through by and you get : the product rule, derived from logs.

When to Use Logarithmic Differentiation

• The variable is in the base and the exponent: , , . Here it’s essentially required.
• A long product or quotient, especially with powers and roots, like Example 2.
• Not for simple functions. For , the product rule is quicker than taking logs.

Common Mistakes

• Forgetting to multiply by at the end. Step 3 gives ; the derivative is times that.
• Writing . It’s , because depends on .
• Inventing log rules. does not split into . Only products, quotients and powers expand.
• Using the power rule on . is only half of the answer.
• Leaving in the final answer. Replace it with the original function.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Differentiate .

Problem 2. Differentiate .

Problem 3. Differentiate .

Problem 4. Differentiate .

Problem 5. Find the slope of at .

That’s the last of the differentiation techniques in this chapter. What’s left is putting them to work: higher-order derivatives, and finding tangent lines to any curve you can now differentiate.