Implicit Differentiation and Hyperbolic Functions

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Implicit Differentiation and Hyperbolic Functions

• Implicit DifferentiationHyperbolic FunctionsThe
  Number e as a Limit

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Implicit Functions
Implicitly Defined Functions
A function f whose values y depend on the
variable x according to an equation in terms of
x and y is said to be implicitly defined by
the equation in question.
Sometimes, but not always, it is possible to
solve y in terms of x from the given
equation. Then one gets an explicit expression
for the function in question.
Examples
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2
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Implicit Differentiation (1)
Differentiate Equations by the Chain Rule
Idea
Examples
1
2
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Implicit Differentiation (2)
Problem
Solution
Equation of the tangent line is y-1-9/13(x-3)
or y-9x/1340/13
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Implicit Differentiation (3)
Problem
B
x
P
y
w
A
The solution uses implicit differentiation and
follows on the next slide.
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Implicit Differentiation (4)
Distance are decreasing, hence the derivatives
are negative
Problem
Solution
Do the airplanes crash?
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Hyperbolic Functions
Definition
tanh
cosh and sinh
The graph of the cosh function is a catenary
curve. It represents a freely hanging cable.
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Hyperbolic Identities
Formulae
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Differentiation of Hyperbolic Functions
Formulae
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Inverse Hyperbolic Functions
Formulae
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Derivatives of Inverse Hyperbolic Functions
Formulae
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The Number e as a Limit