AP Calculus AB and BC

Derivative of e^3x: Answer, Chain Rule, Mistakes

Q: What is the derivative of e^(kx)?

\frac{d}{dx}e^{kx} = ke^{kx}. The exponential returns itself and the chain rule multiplies by the inner derivative k. With k = 3 that is 3e^{3x}.

Q: Why doesn't the exponent decrease like in the power rule?

Because the base is the constant e, not the variable. The power rule \frac{d}{dx}x^{n} = nx^{n-1} needs the variable in the base. In e^{3x} the variable is in the exponent, so the chain rule applies and the exponent stays 3x.

Q: What is the second derivative of e^3x?

Differentiate 3e^{3x} again to get 9e^{3x}. Each derivative multiplies by another factor of 3, so the n-th derivative is 3^{n}e^{3x}.

The derivative of e^(3x) with respect to x is 3e^(3x). The chain rule differentiates the outer exponential, which reproduces e^(3x), then multiplies by the derivative of the inner function 3x, which is 3. The general pattern is that e^(kx) differentiates to k times e^(kx).

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

How to differentiate e^3x

The exponential $e^{u}$ is special: differentiating the outer function reproduces it unchanged. The chain rule then multiplies by the derivative of the exponent.

\frac{d}{dx}e^{u} = e^{u}\cdot\frac{du}{dx}

Here the inner function is $u = 3x$ , whose derivative is $3$ .

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

The same steps give the general linear-exponent rule, worth carrying as one fact.

\frac{d}{dx}e^{kx} = ke^{kx}

Why the 3 appears

Tripling the exponent makes the function climb three times as fast, so every slope is three times the height. That is the factor of $3$ ; it is the actual growth rate, not bookkeeping.

Run backward, the same $3$ becomes a division: $\int e^{3x}\,dx = \frac{1}{3}e^{3x} + C$ . A factor multiplies going forward and divides coming back.

Where the derivative of e^3x shows up on the AP exam

The exponential derivative is Topic 2.7 and the chain rule is Topic 3.1, both on AB and BC. An exponential with a linear inner function is one of the first chain rule examples the College Board uses, and Unit 3 is weighted 5 to 10 percent on each exam.

Do not confuse it with the constant multiple $\frac{d}{dx}(3e^{x}) = 3e^{x}$ . That $3$ is a coefficient out front, while in $e^{3x}$ the $3$ lives in the exponent and enters through the chain rule.

Common mistakes with the derivative of e^3x

Answering $e^{3x}$ and dropping the chain rule factor. This is the most common error; the inner derivative $3$ has to appear.
Answering $3xe^{3x-1}$ by misapplying the power rule. The power rule is for a variable base like $x^{3}$ , not a constant base $e$ raised to a variable.
Answering $3e^{3x-1}$ by lowering the exponent. The exponent of $e$ does not drop; only its coefficient comes out front.

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 e^(kx)?

$\frac{d}{dx}e^{kx} = ke^{kx}$ . The exponential returns itself and the chain rule multiplies by the inner derivative $k$ . With $k = 3$ that is $3e^{3x}$ .

Why doesn't the exponent decrease like in the power rule?

Because the base is the constant $e$ , not the variable. The power rule $\frac{d}{dx}x^{n} = nx^{n-1}$ needs the variable in the base. In $e^{3x}$ the variable is in the exponent, so the chain rule applies and the exponent stays $3x$ .

What is the second derivative of e^3x?

Differentiate $3e^{3x}$ again to get $9e^{3x}$ . Each derivative multiplies by another factor of $3$ , so the $n$ -th derivative is $3^{n}e^{3x}$ .