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Calculus I · Unit 2A · lesson

The Derivative of the Natural Exponential

Concept

Learning objectives

Differentiate exe^x; interpret exponential self-proportional growth.

Why exe^x Is the Natural Exponential

Explanation

Before the formulas

The formulas in The Derivative of the Natural Exponential are easiest to remember when connected to graph behavior. Trigonometric derivatives encode phase and sign patterns, exponential derivatives encode proportional growth, and logarithmic derivatives encode reciprocal sensitivity. The formulas are not unrelated entries in a table.

Keep domain and units visible. Trigonometric derivative formulas use radians. Logarithms require positive real inputs unless a different domain has been explicitly introduced. General exponential and logarithmic bases contribute constants such as ln⁡a\ln a. These details are small on the page and decisive in a correct solution.

For y=e x, the tangent slope at every input equals the function value there. At x=0, both the height and the slope are 1.
Read this graph as text

The special base e has slope equal to height. For y=e x , the tangent slope at every input equals the function value there. At x=0 , both the height and the slope are 1 . The tangent line at (0,1) has slope 1 , matching the exponential's height. At every other input, the same phenomenon persists: the derivative of e x is e x . This is why e is the natural base for continuous growth models and calculus.

The visual uses labeled positions, solid and dashed line styles, and written descriptions so the special base e has slope equal to height does not depend on color.

Why it matters: The graph should convey why base e is mathematically distinguished without pretending that the single tangent at zero proves the derivative formula everywhere. The surrounding prose should connect the visual to the limit definition and to continuous proportional growth.

Visual study

For y=e x, the tangent slope at every input equals the function value there. At x=0, both the height and the slope are 1.

Explanation

The function whose slope matches its height

Most functions change shape when differentiated. The natural exponential is exceptional: at every input, the tangent slope equals the function value. A height of 55 comes with slope 55; a height of 1/21/2 comes with slope 1/21/2.

This self-reproducing behavior is why exe^x appears in continuous growth, radioactive decay, cooling, finance, and differential equations. The same rule describes growth and decay; the sign in the exponent determines the direction.

Most functions change at a rate different from their current size. The natural exponential is special because it reproduces itself under differentiation:

ddxex=ex.\frac{d}{dx}e^x=e^x.

That property makes exe^x the natural language of continuous growth and decay.

Whenever the rate of change is proportional to the amount present, exponential functions appear. Populations, radioactive samples, continuously compounded balances, and idealized drug elimination all share this mathematical skeleton even though their physical stories differ.

The number ee is the unique positive base whose exponential function has slope equal to height everywhere.

Theorem

Derivative of the natural exponential

ddxex=ex.\boxed{\frac{d}{dx}e^x=e^x}.

From the definition,

ex+h−exh=exeh−1h.\frac{e^{x+h}-e^x}{h}=e^x\frac{e^h-1}{h}.

The base ee is characterized by

lim⁡h→0eh−1h=1.\lim_{h\to0}\frac{e^h-1}{h}=1.

Therefore the derivative is exe^x.

Guided walkthrough

Differentiate a polynomial-exponential product

Differentiate

f(x)=x2ex.f(x)=x^2e^x.
Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Concept

Rate proportional to amount

If P(t)=CektP(t)=Ce^{kt}, then P′(t)=kP(t)P'(t)=kP(t). The quantity changes at a rate proportional to its current amount. This pattern appears in population models, continuous interest, cooling approximations, and radioactive decay.

Interactive checkexp-e-01

Differentiate xexxe^x.

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Show hint

This is a product of x and e^x.

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Application

Continuous bacterial growth

A culture is modeled by

P(t)=800e0.35t.P(t)=800e^{0.35t}.

Then

P′(t)=0.35P(t).P'(t)=0.35P(t).

The culture grows at 35%35\% of its current size per hour. When P=2000P=2000, the instantaneous growth rate is 700700 cells per hour, regardless of which time produced that population. Exponential models tie rate directly to current amount.

Optional advanced note

Characterizing the exponential by its derivative

The differential equation y′=yy'=y with initial condition y(0)=1y(0)=1 has the unique solution y=exy=e^x. In analysis, this can be used as a definition of the natural exponential. The function is not merely one convenient growth model; it is the unique function whose local proportional growth rate is constantly one.

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?

Interactive checkexp-extra-01

Differentiate e2xe^{2x}.

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Show hint

Chain rule: multiply by the derivative of 2x2x.

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Source & rights

Original instruction with traceable references.

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