Section 3.9
Derivatives of Hyperbolic Functions
Hyperbolic functions define the shape of hanging cables (catenaries). Their derivatives mirror trigonometry, but with a twist.
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The Formulas
The functions and are defined using and . This leads to a beautiful symmetry in their derivatives.
Sinh to Cosh
Just like sine went to cosine!
Cosh to Sinh
WAIT! No negative sign?
Correct. Unlike cosine, the derivative of cosh is positive sinh.
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Worked Example
Differentiate .
Note: We use the Product Rule here.
Step 1: Identify Parts
Step 2: Product Rule ()
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More Examples
Quotient Rule with Hyperbolics
Differentiate .
Identify components:
Apply Quotient Rule:
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Interactive Visualization
The Catenary Curve
The Blue curve (cosh) is the shape a hanging chain makes. Its derivative (Red) is the hyperbolic sine.
Original Function f(x)
Tangent Line
Derivative Function f'(x)
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Practice Quiz
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