Derivatives of Natural Logarithmic Functions

The Derivative of

The natural logarithm, written , is one of the most important functions in calculus. Its derivative has an unusually simple form:

The key idea is that the derivative of is the reciprocal of its input. This rule is valid wherever is defined, so the input must be positive.

For example, if , then:

Notice that the derivative is not another logarithm. Differentiating removes the logarithm and produces a rational function.

Why the Formula Makes Sense

The natural logarithm is the inverse of the exponential function. In other words, if , then .

Differentiate both sides with respect to :

Since , we have:

Solving for gives:

Because , this becomes:

That is why the derivative of the natural logarithm is .

The General Rule:

The basic formula becomes much more useful when the input to the logarithm is itself a function of . This is a direct application of the chain rule.

In words: take 1 over the inside function, then multiply by the derivative of the inside function.

This is exactly the same pattern as the chain rule for other composite functions. The outer function is , and its derivative is .

A Step-by-Step Method

1. Identify the inside function. Find what is contained inside the natural logarithm.
2. Write 1 over the inside function. This comes from the derivative of .
3. Multiply by the derivative of the inside function.
4. Simplify. Look for factors that cancel or expressions that can be rewritten more clearly.

Worked Examples

An Important Shortcut:

When the argument of the natural logarithm is a positive differentiable function, the rule can be remembered compactly as:

This form is especially useful when the inside function is complicated. Instead of explicitly thinking about "outside" and "inside" functions every time, you can recognize the pattern immediately.

For instance, if , then:

The structure is always the same: derivative of the inside divided by the inside.

When the Input Is Not Simply

A common mistake is to assume that the derivative of every natural logarithm is simply . That is only true when the input itself is .

Compare these two examples:

In the second example, the extra factor appears because the derivative of the inside function is .

Common Mistakes

AP Calculus Connection: Logarithmic Differentiation

The derivative rule for becomes particularly powerful when used for logarithmic differentiation. This technique is useful when a function contains products, quotients, or variable exponents that would be difficult to differentiate directly.

For example, consider:

Take the natural logarithm of both sides:

Using the logarithm power rule:

Differentiate both sides implicitly:

Finally, multiply by :

The important AP Calculus idea is that the derivative of provides a simple way to turn complicated multiplication, division, and exponent structures into sums and differences before differentiating.

Practice Problems