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

The proof: rewrite the base as a power of e

The $a^x$ rule is not a separate fact to memorize. Every positive number is a power of $e$, since $a = e^{\ln a}$. Raise both sides to the $x$ and the exponents multiply, which turns any exponential into a base-$e$ exponential with a linear inner function.

Now differentiate $e^{x \ln a}$ with the chain rule. The outside is $e^u$, whose derivative is $e^u$; the inside is $u = x \ln a$, and because $\ln a$ is a constant, $\frac{du}{dx} = \ln a$.

The base comes back down unchanged and a constant factor of $\ln a$ drops out front. That factor is the whole story: it is why every exponential except one carries an extra number when you differentiate it. The lone exception is base $e$. Set $a = e$ and the factor becomes $\ln e = 1$, so $a^x \ln a$ collapses to $e^x \cdot 1 = e^x$. That is the real reason $e^x$ is its own derivative, and the reason $e$ is the natural base to build on.

a^x is not x^a: the base-versus-exponent trap

The most common wrong turn is reaching for the power rule. The power rule $\frac{d}{dx}\, x^n = n x^{n-1}$ applies when the variable is the base and the exponent is a constant. In $a^x$ the roles are swapped: the base is constant and the variable is the exponent, so the power rule does not apply at all. Read the position of $x$ before you pick a rule.

The $x^x$ row is a reminder that when the variable appears in both places, neither rule alone works and you take $\ln$ of both sides first. That case is past what $a^x$ needs, but it shows why naming the base and the exponent out loud, before writing anything, is worth the two seconds it costs.

Where a^x shows up on the AP exam

The CED lists the derivatives you are expected to know cold in Topic 2.7, "Derivatives of cos x, sin x, e^x, and ln x." Notice $a^x$ is not on that list. The exam expects you to build it from two things that are: the $e^x$ rule in Topic 2.7 and the chain rule in Topic 3.1. Rewriting $a^x$ as $e^{x \ln a}$ is the bridge between them.

Unit 2 is 10-15% of the AB exam and 5-10% of BC; Unit 3, which owns the chain rule, is 5-10% of both. Topic 3.5, "Selecting Procedures for Calculating Derivatives," is where the exam checks that you recognize an exponential base and reach for $a^x \ln a$ rather than the power rule.

In practice $a^x$ arrives most often inside a composite such as $a^{g(x)}$, or in a table-of-values problem that hands you values of a function $f$ and asks for the derivative of $2^{f(x)}$ at a point. Both reduce to the same rule plus one chain-rule factor.

Common mistakes

• Using the power rule: writing $\frac{d}{dx}\, a^x = x\, a^{x-1}$. This misapplies the power rule. The power rule is for $x^a$, where the variable sits in the base, and gives $a x^{a-1}$; here the variable is in the exponent, so the exponential rule applies instead.
• Dropping the $\ln a$ factor: writing $\frac{d}{dx}\, a^x = a^x$. That is true only for $a = e$. Every other base leaves a constant factor of $\ln a$ out front.
• Forgetting the inner derivative on a composite: $\frac{d}{dx}\, 3^{x^2}$ is not $3^{x^2} \ln 3$. The chain rule multiplies by $\frac{d}{dx}\, x^2 = 2x$.
• Confusing $\ln a$ with $\ln x$: the factor is $\ln$ of the constant base, a fixed number, not a function of $x$. For $2^x$ it is the constant $\ln 2 \approx 0.693$.

Two worked composites with the chain rule

Both use the same composite form. Writing $F(x) = a^{g(x)}$ gives $F'(x) = a^{g(x)} \ln a \cdot g'(x)$: the base rule times the derivative of the exponent.

Example 1. Differentiate $3^{x^2}$. Here $F(x) = 3^{x^2}$, the base is $3$, the inner function is $g(x) = x^2$, and $g'(x) = 2x$.

Example 2. Differentiate $5^{\sin x}$. Here $F(x) = 5^{\sin x}$, the base is $5$, the inner function is $g(x) = \sin x$, and $g'(x) = \cos x$.

In each case the base stays put, $\ln$ of the base rides in front as a constant, and the chain rule tacks on the derivative of the exponent. If you can reproduce those three moves in order, the general exponential is no harder than $e^x$.