AP Calculus AB and BC

Power Rule vs Exponential Rule

Q: What is the derivative of x to the x?

It is x^x(\ln x + 1), found by logarithmic differentiation, because neither the power rule nor the exponential rule applies.

Use the power rule when the variable is the base and the exponent is a constant, as in x cubed. Use the exponential rule when the variable is the exponent and the base is a constant, as in two to the x. When the variable is in both places, neither rule works and you need logarithmic differentiation.

Power rule

Use when: The variable is the base and the exponent is a fixed number, including roots and reciprocals.

Exponential rule

Use when: The base is a fixed number and the variable sits in the exponent.

Side by side

	Power rule	Exponential rule
Form	$x^n$	$a^x$
Derivative	$n x^{n-1}$	$a^x \ln a$
Variable is	The base	The exponent
Example	$x^5 \to 5x^4$	$2^x \to 2^x \ln 2$

The power rule is wider than it looks, because roots and reciprocals are powers in disguise: $\sqrt{x}$ is $x^{1/2}$ and $\frac{1}{x^3}$ is $x^{-3}$ . Rewriting first turns both into one-step problems.

The natural exponential is the clean special case. Since $\ln e = 1$ , the derivative of $e^x$ is just $e^x$ , which is why it shows up everywhere.

When the variable is in both places

For something like $x^x$ neither rule applies. Take the natural log of both sides and differentiate implicitly instead.

Frequently asked questions

What is the derivative of x to the x?

It is $x^x(\ln x + 1)$ , found by logarithmic differentiation, because neither the power rule nor the exponential rule applies.

In the CED: Unit 2: Defining the Derivative, Unit 3: Chain Rule, Implicit, and Inverses