The General Exponential Function
You already know the special exponential function . Its derivative is unusually simple because . But exponential functions can have any positive base other than 1, such as .
For a general exponential function , the derivative is:
The important new factor is . This is the natural logarithm of the base. When the base is , we have , so the familiar rule for is recovered automatically.
Why Does Appear?
The number is not arbitrary. Any exponential function with base can be rewritten using the natural exponential function:
Now differentiate using the chain rule. Since the derivative of is itself, the extra factor that appears is exactly :
This is the underlying reason the general exponential derivative rule works.
The Chain Rule Version
On the AP exam, you will often see a function such as , where the exponent is itself a function of . In that situation, apply the chain rule:
Think of it in three pieces: keep the original exponential expression, multiply by , and then multiply by the derivative of the exponent.
Worked Examples: Exponential Functions
Differentiate .
Differentiate .
Differentiate .
The General Logarithmic Function
The logarithmic rule has a similar base-dependent factor. For the natural logarithm, you know:
For a logarithm with a general base , the derivative becomes:
Again, the natural logarithm appears. The denominator contains because changing the logarithm's base changes the scale of the function.
Why Does Appear?
Use the change-of-base formula:
Since is a constant with respect to , differentiate:
So the general logarithmic derivative rule is simply the familiar natural-log rule divided by .
The Chain Rule for General Logarithms
If the argument of the logarithm is itself a function , then the chain rule gives:
The pattern is worth memorizing: derivative of the inside divided by the inside, then divided by .
Worked Examples: Logarithmic Functions
Differentiate .
Differentiate .
Differentiate and compare it with .
Exponential vs. Logarithmic Derivatives
The two general rules are closely related:
For exponentials, appears as a multiplicative factor. For logarithms, appears in the denominator.
A useful way to remember the distinction is:
- Exponential: keep the exponential and multiply by .
- Logarithm: divide by the input and also divide by .
Important Domain Restrictions
For , the base must satisfy . A base of 1 does not produce a meaningful logarithm or a nonconstant exponential function.
For logarithms, the argument must be positive. Therefore, if , then:
For example, is only defined where . The derivative formula is not a substitute for checking the domain of the original function.
Common Mistakes
A common error is to write for the derivative of . The correct derivative is . The factor disappears only when the base is .
For exponentials, is multiplied. For logarithms, is in the denominator. Mixing these up changes the derivative completely.
For , the derivative is not simply . You must multiply by . Similarly, requires the derivative of its inside function.
The real-valued function requires . Even though the algebraic derivative expression may look valid elsewhere, the original logarithmic function is not defined there.
Practice Problems
Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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