f x derivative offix
f i derivative offax
x
FI 2x a y 2 2
y X 261 1
output
rivative of
input
y withrespect to x
Gf x at
9 1,2 2121 2
differentiable derivative exists
domain or
a specificinput
Possibilities ofnot differentiable
1 no point
an input
2 one sided limits aren'tequal
r
F lx
Ixty
f x
x
x
xo
x
O
1 x co
1 x O
f o DNE bk slopes aren'tmatching up
Y
If fax is not continous
function is not continous
so
f 1 DNG
an x value then it
is not differentiable
two linearpieces with different slopes meeting up
Coger
F FA is continous
x
3
but not differentable
y
n fix
XE O
3 12
1 x 0
a cont
x 0
Flo I
tim X 1 1
o
2
lip03
B differentiable
2
1
1
X 0
take derivative for each function at x o to get fix
Fx
2x
XO
2
6
2107 0 E 610 2
Iim F x 2107 0
x
O
7 41 6
o
2
0
x
does not match up
2
F'd
Dre
2
if a function is differentiable it must be continous
Fx 4
Fx 0
4
2
glx 2 1
It
glx 2
L
hx y
u
nextligandhad
ImaxI tip
it
9 29
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yaimggxtnpI
3 3 4 3xtith
limmo3xshtpxhth'slim 35 3 4 42 3
2
no
Pffatintian are s
a n xn l
Fx
1 fix X
Fx
2
2 x2 2x
x X
g
31
gx 3
3
2
32
3 hx 5 4 2
2
3 1
n x 28 4 13
3
Flyworksforadditionesubtraction
XI
Ext I
4 glx 2 4 1 2 2 2x
x 4 2
5 FIXER
Fx
g
6 gx
gx
no negative or fractional exponents as finalanswer
3
1 x 43
4
3
3
xn
7 y 3
IF
can notdo this
n xn
when writting a piecewise function for F x first see if F x is
continous at that point so you know weather or not to include it
ex
fix
3
x o
x so
f x is continous ok limfox Flo
Fx
is
not continous ble f o one so in the piecewise
function don't write x so
only write x
co
corner points on a graph do not exist as the slopes from eachside
are not equal
Normal line negative reciprocal of slope
horizontal tangents derivative o
whenvelocity changessigns it means object turns around so to
find this make a number line
slowingdown velocity a acceleration are different signs
to see if a function in inc or decat a certain point find the velocity or
function is t a vice versa
derivative at that point a if its