Derivatives
The derivative is the slope of the tangent line
Average slope
Instantaneousslope
Δ
Δ
lim
→
Δ
Δ
Defn. 1 (lots of distributive multiplication):
d
d
′
∶
Δ
Δ
lim
→
Defn. 2 (lots of factoring):
′
@
d
d
∶
lim
→
′
Notating a derivative of a function
Review of functions
a name of Platonic ideal of a particular function
an arbitrary yet specific and particular value on the real number line
the particular value that is associated with by the function ; can be expressed as an
explicit association rule computing formula in terms of
Derivatives
′ a.k.a.
′
a name of Platonic ideal of derivative of function named
the particular value that is associated with by the function ′; can be
expressed as an explicit association rule computing formula in terms of
a.k.a.
Abuse of notation
where you see this, you can replace it with the association
rule formula (in terms of ) corresponding to the derivative
of the function whose association rule is named or written
within the brackets to which ′
is applied.
expressionintermsof
d
d
expressionintermsof
The symbols ′
and
where you see this, you can replace it with the association
rule formula (in terms of ) corresponding to the derivative
of the function whose association rule is named or written
is applied.
within the brackets to which
are distinct from the symbols ′ and
. Abuse notation by omitting
the distinguishing mark REPR.
Page 1 of 4
DAVID LIAO.COM
Derivatives
Differentiation rules (valid provided that ′
Derivative of a constant
and ′
exist)
Sum (or difference) rule
Power rule
0
Derivative of a constant
times a function
′
2
Usually written
′
′
d
d
Chain rule for derivative of a composite function
d
d
d
d
Shorthand:
Page 2 of 4
d
d
d d
⋅
d d
⋅
d
d
1. Use order of operations to identify sequence
from innermost to outermost layer. Ask, “What
do I do first to ? Then what do I do next?”
Continue until you have enumerated all
operations. Distinguish the different layers
using different colors.
1 ⋅ spaceformorewriting.
2. Write stuff
3. Identify the currently outermost yet-to-bedifferentiated layer and regard everything else
in deeper layers as a single abstract variable.
4. Differentiate the currently outermost yet-to-bedifferentiated layer with respect to everything
else in deeper layers regarded as a single
abstract variable.
5. Multiply the until-now formed product of factors
by the expression obtained in step 4.
6. What was the single abstract variable with
which you “with respected to” in step 4? Was it
itself? If not, repeat steps 3-6. If yes, stop.
DAVID LIAO.COM
Derivatives
Product rule
d
d
lim
d
d
d
Δ
Δ
→
d
d
d
lim
→
lim
d
d
→
Δ
Area Beam
Area Pillar
Δ
Δ ⋅
Δ
⋅Δ
d
d
′
Quotient rule
′
Derivatives do not always exist
Page 3 of 4
DAVID LIAO.COM
Derivatives
Higher-order derivatives
Notation
Lagrange
Original function
Leibniz
d
d
d
d
d
d
d
d
′
First derivative
′
′′′
⋮
Second derivative
⋮
′′
d
d
Explicit differentiation
vs.
Implicit differentiation
3
3
3
⋅
3
⋅
1
3
1⋅
′
⋅ ′
3′
0
⋅ ′
′
Page 4 of 4
DAVID LIAO.COM