AP Calculus AB and BC

Derivative of 3^x: Answer, Proof, and Mistakes

Q: What is the derivative of 3^{x}?

It is 3^{x}\ln 3. The factor \ln 3\approx 1.0986 comes from writing 3^{x} as e^{x\ln 3}.

Q: What is the derivative of a^{x} in general?

For any a>0 it is a^{x}\ln a. Base e is the case where \ln a = 1, so the derivative reproduces the function.

The derivative of 3^x is 3^x ln 3, roughly 1.0986 times 3^x. In prime notation, if f(x) = 3^x then f'(x) = 3^x ln 3. Rewrite 3^x as e^(x ln 3) and the chain rule multiplies by the inner derivative ln 3, which is the general rule that a^x differentiates to a^x ln a.

\frac{d}{dx}\left[3^x\right] = 3^x\ln 3

How to differentiate 3^x

Convert the base to $e$ first. Since $3 = e^{\ln 3}$ , the function is $3^{x} = e^{x\ln 3}$ , an exponential with the constant $\ln 3$ inside. Now the chain rule applies, with inner derivative $\ln 3$ .

\frac{d}{dx}3^{x} = \frac{d}{dx}e^{x\ln 3} = e^{x\ln 3}\cdot\ln 3 = 3^{x}\ln 3

Nothing about the base $3$ mattered except that it is positive, so the same argument gives the general exponential rule.

\frac{d}{dx}a^{x} = a^{x}\ln a \quad (a>0)

Why base e is the special case

The derivative of $3^{x}$ is a fixed multiple of the function itself, and $\ln 3\approx 1.0986$ is that multiplier, so the slope is always about $9.9$ percent larger than the height. Base $e$ is the one case where $\ln a = 1$ and the derivative equals the function exactly.

f'(0) = 3^{0}\ln 3 = \ln 3 \approx 1.0986

That number is the slope of $y = 3^{x}$ as it crosses the $y$ -axis at height $1$ , and comparing it with the slope $1$ of $y = e^{x}$ at the same height is the cleanest way to see why $e$ is the natural base for calculus.

Common mistakes with the derivative of 3^x

Using the power rule and answering $x\cdot 3^{x-1}$ . The power rule needs a variable base and a constant exponent; here it is the other way around.
Answering $3^{x}$ with no logarithm. Only base $e$ has a derivative equal to itself.
Writing $3^{x}\ln x$ . The logarithm takes the base $3$ , a constant, not the variable.
Turning the rule upside down as $\frac{3^{x}}{\ln 3}$ , which is the antiderivative, not the derivative.

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 $3^{x}$ ?

It is $3^{x}\ln 3$ . The factor $\ln 3\approx 1.0986$ comes from writing $3^{x}$ as $e^{x\ln 3}$ .

Why does $\ln 3$ appear in the answer?

Because the chain rule multiplies by the derivative of the exponent in $e^{x\ln 3}$ , and that exponent is $x\ln 3$ , whose derivative is the constant $\ln 3$ .

What is the derivative of $a^{x}$ in general?

For any $a>0$ it is $a^{x}\ln a$ . Base $e$ is the case where $\ln a = 1$ , so the derivative reproduces the function.