Calculus I · Unit 2A · lesson

Derivatives of General Exponential Functions

Concept

Learning objectives

Differentiate $a^x$ and explain the role of $\ln a$ .

Bases Other Than $e$

Explanation

Before the formulas

In Derivatives of General Exponential Functions, begin by asking what the original function does. Where does it rise, fall, flatten, or grow in proportion to itself? The derivative formula should match that qualitative behavior. A missing minus sign in the cosine derivative, for example, contradicts the fact that cosine decreases immediately to the right of zero.

When several special functions appear together, separate recognition from computation. Identify each basic derivative, note any composition requiring the chain rule, and then combine the pieces. A short verbal plan keeps a crowded formula from becoming a guessing contest.

Explanation

Other bases carry a growth-rate constant

A function $a^x$ grows by the same percentage for equal input changes, but its instantaneous relative growth rate is $\ln a$ , not always one. Writing $a^x=e^{x\ln a}$ exposes the chain rule and explains the derivative $a^x\ln a$ .

For $0<a<1$ , $\ln a<0$ , so the derivative is negative. The formula therefore encodes the difference between exponential growth and exponential decay automatically.

A base such as $2$ , $10$ , or $1.03$ changes the scale of exponential growth. The factor $\ln a$ in

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

measures how aggressive that base is. Bases above $1$ give positive rates; bases between $0$ and $1$ give negative rates.

The formula becomes especially useful when the exponent is itself a function. Then the exponential rule and chain rule work together, separating the current amount from the rate at which the exponent changes.

For $a>0$ , write

a^x=e^{x\ln a}.

The chain rule, developed fully in the next section, gives

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

The factor $\ln a$ measures how rapidly the chosen base grows relative to $e$ .

Guided walkthrough

Differentiate a base-2 exponential

Differentiate $f(x)=3\cdot2^x$ .

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

Exponential decay base

For $g(t)=5(0.8)^t$ ,

g'(t)=5(0.8)^t\ln(0.8).

Since $\ln(0.8)<0$ , the derivative is negative, matching decay.

Common mistake

The power rule does not apply to $2^x$ . In $x^n$ , the variable is the base. In $a^x$ , the variable is the exponent. Those are different structures.

Application

A depreciating device

A device worth $V(t)=2400(0.82)^t$ dollars after $t$ years has derivative

V'(t)=2400(0.82)^t\ln(0.82).

At purchase, $V'(0)=2400\ln(0.82)\approx-476$ dollars per year. The derivative becomes less negative over time because the same percentage decline acts on a smaller remaining value.

After the explanation

Use the section idea

Reading lens

Connect each derivative formula to the function's characteristic growth, periodicity, or inverse relationship.

Mental model

Special-function rules preserve recognizable shapes while scaling them by a function-specific factor.

Decision

Identify the function family first, then check whether a composition requires the chain rule too.

Common trap

Using a power rule on an exponential or forgetting base and domain conditions for logarithms.

Check yourself

Can you distinguish a power, exponential, logarithmic, and trigonometric derivative at a glance?

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Bases Other Than eee

Original instruction with traceable references.

Bases Other Than $e$