Calculus I · 2A · lesson
The Derivative of the Natural Exponential
Learn the derivative of the natural exponential with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Trigonometric, exponential, and logarithmic functionsWhat this section is building
Learn the derivative of the natural exponential with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Special-function rules preserve recognizable shapes while scaling them by a function-specific factor.
Identify the function family first, then check whether a composition requires the chain rule too.
Using a power rule on an exponential or forgetting base and domain conditions for logarithms.
Learning objectives
Differentiate ; interpret exponential self-proportional growth.
Why Is the Natural Exponential
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 . These details are small on the page and decisive in a correct solution.
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.
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 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 comes with slope ; a height of comes with slope .
This self-reproducing behavior is why 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:
That property makes 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 is the unique positive base whose exponential function has slope equal to height everywhere.
Derivative of the natural exponential
From the definition,
The base is characterized by
Therefore the derivative is .
Differentiate a polynomial-exponential product
Differentiate
Worked solution
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Rate proportional to amount
If , then . The quantity changes at a rate proportional to its current amount. This pattern appears in population models, continuous interest, cooling approximations, and radioactive decay.
exp-e-01Differentiate .
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Show hint
This is a product of x and e^x.
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Continuous bacterial growth
A culture is modeled by
Then
The culture grows at of its current size per hour. When , the instantaneous growth rate is cells per hour, regardless of which time produced that population. Exponential models tie rate directly to current amount.
Characterizing the exponential by its derivative
The differential equation with initial condition has the unique solution . 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.
exp-extra-01Differentiate .
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Show hint
Chain rule: multiply by the derivative of .
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