Section 3.7
Exponentials and Logarithms
These functions are unique in calculus. One is its own derivative, and the other fills the gap in the Power Rule.
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Calculus Superstars
The Natural Exponential
The only function (scaled by a constant) whose slope equals its height everywhere. It is the fixed point of differentiation.
The Natural Logarithm
This fills the "hole" in the Power Rule for . This simple derivative links logarithms to hyperbolas!
General Bases (Optional but Good to Know)
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Worked Example
Differentiate .
Apply Limits Linearly
Substitute Known Formulas
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More Examples
Product Rule with
Differentiate .
Identify parts:
Apply :
Quotient Rule with
Differentiate .
Identify parts:
Apply :
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Interactive Visualization
The Magic of e^x
Look closely at the graph. The Blue curve (function) and Red dashed curve (derivative) overlap PERFECTLY. The slope at any point equals the y-value.
Original Function f(x)
Tangent Line
Derivative Function f'(x)
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Practice Quiz
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