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Differentiation topics
Difference quotient for f at x = a
𝑓 (𝑎+h )− 𝑓 (𝑎)h
Definition of derivative
limh→0
𝑓 (𝑥+h )− 𝑓 (𝑥)h
If f is differentiable at x = a…
Then f is continuous at x = a
Rolle’s TheoremIf f is continuous on ,
differentiable on , and then…
For some c in ,
Mean Value Theorem of Derivatives
If f is continuous on , differentiable on , then…
For some c in ,
Mean Value Theorem of Derivatives
(as rates of change)If f is continuous on ,
differentiable on , then…
At some point in the interval the
instantaneous rate of change equals the
average rate of change
If f is differentiable at x = a…
limh→0
𝑓 (𝑎+h )− 𝑓 (𝑎)h
∃
𝑑𝑑𝑥
(k · 𝑓 )
k · 𝑓 ′
𝑑𝑑𝑥
( 𝑓 ±𝑔 )=¿
𝑓 ′ ±𝑔 ′
=
Product Rule
=
Quotient Rule
=
Chain rule
=
1𝑓 ′ (𝑦)
=
𝑎𝑥𝑎−1
=
cos 𝑥
=
−𝑠𝑖𝑛𝑥
=
=
=
𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥
=
-
=
1
√1−𝑥2
=
−1
√1−𝑥2
=
1
1+𝑥2
=
−1
1+𝑥2
=
1
|𝑥|√𝑥2−1
=
−1
|𝑥|√𝑥2−1
=
𝑒𝑥
=
𝑎𝑥 𝑙𝑛𝑎
=
1𝑥
=
1𝑢𝑙𝑛𝑎
𝑑𝑢𝑑𝑥
Average rate of change of f on
𝑓 (𝑏 )− 𝑓 (𝑎)𝑏−𝑎
Instantaneous rate of change
of f at
𝑓 ′(𝑎)
, then…
The graph of f has a horizontal tangent at
𝑑𝑥𝑑 𝑦
=0
The graph of f has a vertical
tangent
Critical points of f
•At the endpoints of the domain•Where f’ does not exist•Where f’ = 0
𝑓 ′ (𝑥 )>0 for𝑎<𝑥<𝑏
𝑓 ′ (𝑥 )<0 for𝑎<𝑥<𝑏
𝑓 ′ ′ (𝑥 )>0 for 𝑎<𝑥<𝑏
𝑓 ′ ′ (𝑥 )<0 for 𝑎<𝑥<𝑏
𝑓 ′ (𝑎)=0𝑎𝑛𝑑 𝑓 ′ ′ (𝑎)<0
f has a maximum at x = a (2nd derivative test)
𝑓 ′ (𝑎)=0𝑎𝑛𝑑 𝑓 ′ ′ (𝑎)>0
f has a minimum at x = a (2nd derivative test)
What is a point of inflection?
A point where the graph of f changes
concavity
Velocity
, where s(t) = position
Speed
|𝑣(𝑡)|
Acceleration (in terms of velocity)
, where v(t) = velocity
Acceleration (in terms of position)
, where s(t) = position
Increasing Speed
Velocity and acceleration have
the same sign
Decreasing Speed
Velocity and acceleration have
opposite signs
An object is at rest when…
𝑣 (𝑡 )=0
