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. .

.