AP Calculus AB and BC

Derivative of 2^x: Answer and Why ln 2 Appears

Q: Why does ln 2 appear?

Because 2^{x} = e^{x\ln 2}, and the chain rule brings down the \ln 2 from the exponent.

Q: How is 2^x different from x^2?

x^{2} has the variable in the base and differentiates to 2x. 2^{x} has it in the exponent and differentiates to 2^{x}\ln 2.

The derivative of 2 to the x is 2 to the x times the natural log of 2. The variable sits in the exponent rather than the base, so this is an exponential function and the power rule does not apply. The ln 2 factor is about 0.693.

\frac{d}{dx}\left[2^{x}\right] = 2^{x}\ln 2

Where ln 2 comes from

Rewrite the base as a power of $e$ , which is the only exponential the chain rule handles directly.

2^{x} = e^{x\ln 2} \implies \frac{d}{dx}2^{x} = e^{x\ln 2}\cdot\ln 2 = 2^{x}\ln 2

The general rule follows the same way, and $e$ is the special case where the extra factor is $1$ .

\frac{d}{dx}a^{x} = a^{x}\ln a

Base against exponent

$x^{2}$ has the variable in the BASE and differentiates to $2x$ by the power rule. $2^{x}$ has it in the EXPONENT and differentiates to $2^{x}\ln 2$ . Mixing them up is the most common error on this topic.

Common mistakes

Answering $x2^{x-1}$ , applying the power rule to an exponential.
Answering $2^{x}$ and forgetting the $\ln 2$ .
Writing $\log 2$ instead of $\ln 2$ . The natural logarithm is what appears.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of 2^x?

It is $2^{x}\ln 2$ .

Why does ln 2 appear?

Because $2^{x} = e^{x\ln 2}$ , and the chain rule brings down the $\ln 2$ from the exponent.

How is 2^x different from x^2?

$x^{2}$ has the variable in the base and differentiates to $2x$ . $2^{x}$ has it in the exponent and differentiates to $2^{x}\ln 2$ .