Title: Calculus 3.1
1Derivatives Differentiability
Great Sand Dunes National Monument, Colorado
2is called the derivative of at .
The derivative of f with respect to x is
3the 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
4Note
dx does not mean d times x !
dy does not mean d times y !
5Note
(except when it is convenient to think of it as
division.)
(except when it is convenient to think of it as
division.)
6Note
(except when it is convenient to treat it that
way.)
7In the future, all will become clear.
8So what is the derivative anyway?
9Find the derivative of the function and graph
both y and y
10A 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.
11To be differentiable, a function must be
continuous and smooth.
Derivatives do not exist at
corner
cusp
discontinuity
vertical tangent
12Most of the functions we study in calculus will
be differentiable.
13 Example
Find and graph both.
14 So if yx3,
15 Ex. Over what interval(s) is each function
differentiable?
Corner/cusp at x 2
Vertical tangent line at x 5
16Differentiability 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.)
17Continuous over all real numbers, but not
differentiable over all real numbers!