Derivatives 9/26/18
Taste,
we defined the derivative of fat a by
five METI . fit this exists)
This gave us the slope of the tangent line to the graph of f at ai
f - feat flake - al.
Renard Another way to write FIN is given by taking a limit as ex-doo instead of as wa.
Write h for Had.
Then
fq-affefhtayal.fr#=flhtNh-frN=frathlyfai .
So
flat ¥now.
HITT i hire. HatN-f .
We can use our techniques of computing limitsto find derivatives and tangent lines.
trample What is the derivative of free K at eel ? What is the equation of the tangent
line to thegraph of f at Hell Iwhich Ia point on the graph.
)
Solution! We plug in the definition.si
f'
g) = thingfHtN-f GMI i tier .
h
. ÷ . .÷"÷ '
=I in It 2h th '
- 1
hoo T-
yintercept is - I
= IIo 2¥
A= hi:O 2 th
= 2.
The tangentline at hell is then
Kettle - Il.
Keli2K - H
y - 1=2 a -2
You can get slope - intercept form Him by rearranging !
y ⇒ a -2¥
Y
±2k - I
Aninportmfpa.nl#The derivative of a
function is itself a function ! that is.
if we have a function f ,the
derivative ,defined by
f-'
ro - the derivative of f at x
is again a function of t.
Example Let's find f'
co for fruit ?
flushing fruhl.fr#h
=him
.
= thing NtUhtth
= fifo 2xhh
= thin,
2x th
= 2x.
This says the slope of the tangent line to the graph of yell at Ka is always 2A.
Easiereump What is free for
fruit?
flu slimfattyhad
= hi:OEtee
I in 3kt 3h -3khoo I
=iinto In
=3 .
This makes sense : the tangent line to a line at any of its points should just be the line itself.
What about giro forfro
-
- or a constant ? By this logic.
we should get zero.
Let's see :
I in NHN - NO I,
no e-i n'Io = o V
A funny derivative
toffee ! Then Hosek as well ! et is the only function f on the whole real line
With the properties that f'
yl i few and folie.
We will explore this derivative later.
Differentiability
afudimfiscwlefdifferent.ie#ataifftaoeeistXIt is differentiable on on open interval if it is differentiable
at every number in that interval-
Non-empty let free IN.
Remember.
'
Heft if xeo- x if X so
.
If
free hino HAI ( uh is positive if his near enough to x )h
= fig etherh
-
-
"n:O ha
=L,
If to,
fine limo
EINENKth is negative if
hisnear enough to x )
hlim - A th ) - I - N
= hoof= Io
- theh
iii.no In= -1
.
Now.
at
Iim Hoth ) - fro) I im froth ) - f loltrot I
h-io-h-hI.itHh Bug Inigo .
FINI6k trio from the h
frontso no o In 4¥!:OIII.then Io .
-
ha=L
= -1.
The left and right limits defining ftol don't agree .so free IN is not differentiable at x -0
.
him " " "
e
This is because of the "
corner" at no . Graphs with corners fail to be differentiable at those corners .
theorem If I is differentiable of a.
then f is continuous at a.
We just saw that the converse is not tree ! fret IN is continuous at KO.
but it snotdifferentiable there
.
higherorderterivafvessometimes people write :
¥ for f'
.
These mean the some thing.
Often,
it is useful to fate derivatives of derivatives,and so on
:
Second order derivative ! IIe . or f"
,the derivative of the function f
'
.
third order derivative : 1¥ .or for the derivative of f
"
.
And so on. . . III. or f
" '.
Physical interpretation . If fotiis the position of something at time t
.
then :
flu position
f'
IN velocity
it:/:: nation
There are lots of mathematical - physical jokes involving third derivatives. .
.
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