Differentiation topics

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