Logarithmic Differentiation

What Logarithmic Differentiation Is For

Some functions are technically differentiable using the rules you already know, but doing so directly can become extremely complicated. A classic example is a function containing several factors multiplied together, such as .

An even more important case occurs when the variable appears in both the base and the exponent, such as . The ordinary power rule does not apply because the exponent is not constant, while the ordinary exponential rule does not apply because the base is not constant.

Logarithmic differentiation solves these problems by taking the natural logarithm of both sides first. The logarithm converts multiplication into addition, division into subtraction, and powers into coefficients. Differentiation then becomes much easier.

The Key Idea

Suppose . Take the natural logarithm of both sides:

The logarithm lets us use the logarithm rules to rewrite the expression. For a product, for example:

Once the expression has been rewritten in this form, differentiate both sides implicitly with respect to . Because , the derivative of the left side is .

Finally, solve for . The original function is then substituted back in at the end.

The Basic Procedure

Take the natural logarithm of both sides. Write .
Use logarithm properties to simplify the right-hand side. Products become sums, quotients become differences, and exponents become coefficients.
Differentiate implicitly. Remember that , so its derivative is .
Solve for . Factor out when necessary.
Substitute the original expression for .
Simplify. Look for a form that is easy to interpret and use.

Worked Examples

Example 1: A Product of Several Factors

Differentiate .

Step 1: Take the natural logarithm of both sides: .

Step 2: Use the product rule for logarithms: .

Step 3: Differentiate both sides: .

Step 4: Multiply by : .

Result: .

Example 2: A Quotient

Differentiate .

Step 1: Take the natural logarithm: .

Step 2: Use the quotient and power properties: .

Step 3: Differentiate: .

Step 4: Multiply through by : .

Result: .

Example 3: Variable Base and Variable Exponent

Differentiate .

Step 1: Take the natural logarithm: .

Step 2: Bring the exponent down using the logarithm power rule: .

Step 3: Differentiate both sides using the product rule: .

Step 4: Multiply by : .

Result: .

Example 4: A More Complicated Product

Differentiate .

Step 1: Take the logarithm of both sides: .

Step 2: Expand the logarithms: .

Step 3: Differentiate term by term: .

Step 4: Multiply by : .

Result: .

When Should You Use Logarithmic Differentiation?

You do not need logarithmic differentiation for every complicated-looking function. It is especially useful when the expression has one or more of these features:

A product containing several factors.
A quotient with complicated expressions in the numerator and denominator.
Several factors raised to different powers.
A variable appearing in both the base and exponent.
A function that becomes much simpler after applying logarithm properties.

The most important special case is a function such as . The ordinary power rule cannot handle this because the exponent is not a constant. Logarithmic differentiation handles it naturally.

Why the Variable-Exponent Case Works

Consider the general function . Taking logarithms gives:

Differentiating produces:

Multiplying both sides by gives the general result:

This formula is worth recognizing, but on the AP exam it is often safer to understand the logarithmic-differentiation process rather than simply memorizing the result.

Common Mistakes

Forgetting that y is a function of x

When differentiating , the derivative is not simply . Because , the chain rule gives .

Forgetting to multiply by y at the end

After logarithmic differentiation, you often obtain an equation containing . You must multiply through by and then substitute the original function back in. Stopping one line too early leaves the derivative incomplete.

Using the wrong logarithm property

Remember that , while . A quotient becomes a difference, not a product.

Applying the power rule to a variable exponent

The power rule requires a constant exponent. For example, cannot be differentiated by simply writing . The exponent itself depends on , so logarithmic differentiation is an appropriate method.

Practice Problems

Question 1

Differentiate .

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Question 2

Differentiate .

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Question 3

Differentiate .

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Question 4 (Variable Exponent)

Differentiate .

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