Calculus 3.1

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Derivatives Differentiability
Great Sand Dunes National Monument, Colorado
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is called the derivative of at .
The derivative of f with respect to x is
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the derivative of f with respect to x
f prime x
or
y prime
the derivative of y with respect to x
or
dee why dee ecks
the derivative of f with respect to x
or
dee eff dee ecks
the derivative of f of x
dee dee ecks uv eff uv ecks
or
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Note
dx does not mean d times x !
dy does not mean d times y !
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Note
(except when it is convenient to think of it as
division.)
(except when it is convenient to think of it as
division.)
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Note
(except when it is convenient to treat it that
way.)
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In the future, all will become clear.
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So what is the derivative anyway?
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Find the derivative of the function and graph
both y and y
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A function is differentiable if it has a
derivative everywhere in its domain. It must be
continuous and smooth. Functions on closed
intervals must have one-sided derivatives defined
at the end points.
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To be differentiable, a function must be
continuous and smooth.
Derivatives do not exist at
corner
cusp
discontinuity
vertical tangent
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Most of the functions we study in calculus will
be differentiable.
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Example
Find and graph both.
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So if yx3,
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Ex. Over what interval(s) is each function
differentiable?

Corner/cusp at x 2
Vertical tangent line at x 5
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Differentiability implies Continuity, but NOT the
other way around!

• ie.- If f is differentiable at x c, then f is
  continuous at x c.
• But the converse is not necessarily trueJust
  because f is continuous at x c, doesnt
  automatically make it differentiable there.
• (Think of the last example.)

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Continuous over all real numbers, but not
differentiable over all real numbers!