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EECM 3714

Lecture 4: Unit 4
Differentiation

Renshaw, Ch
...
4-10,13
04 March 2022

OUTLINE
Renshaw, Ch
...
4-10 & 13
1
...
Differentiation rules
3
...
Examples
5
...

• Look at fig
...
1a and 6
...
When we move from P to Q, we measure the slope of 𝑦 = 𝑎𝑥 + 𝑏 as the
change in y, Dy, divided by the change in x, Dx
...


The diff
...
measure the slope, or rate of change of y as x
changes, between P and Q
• In fig
...
1a the diff
...
is positive because Dy is positive; in
fig
...
1b it is negative because Dy is negative
...
6
...
3

Δ𝑦
Δ𝑥

is the same in both, but the curves are very different
...
quot
...

• Another problem: the diff
...
also varies with distance from P to Q
...
4 to address this problem, we can use the slope of the tangent to the

curve at P as the measure of slope of curve at that point
...
5)
•

Δ𝑦
Δ𝑥

then approaches a limiting value, which

measures slope of tangent at P
...

∆𝑦
∆𝑥→0 ∆𝑥

• Derivative is lim

𝑑𝑦

= 𝑑𝑥

• So if 𝑦 = 𝑓(𝑥) then the slope of the function is
∆𝑦
∆𝑥→0 ∆𝑥

𝑓 ′ 𝑥 = lim

𝑑𝑦

= 𝑑𝑥

RULES OF DIFFERENTIATION
• “Differentiation” means finding derivative of a function
...

• For any function 𝑦 = f(𝑥), we write its derivative as:
either

d𝑦
d𝑥

or f ′ 𝑥

• The notation f′(𝑥) is obviously more compact
...
Power rule

y=x

n

2
...
Additive constant

y = f( x ) + B

4
...
Power rule: if 𝑦 = 𝑥 3 , we have 𝑛 = 3, so

d𝑦
d𝑥

= 3𝑥 3−1 = 3𝑥 2

2a
...
Multiplicative constant:if y = Ax, then

𝑑𝑦
𝑑𝑥

=𝐴

3
...
Sum or difference: 𝑦 = 𝑥 3 + 𝑥 2 ,

d𝑦
d𝑥

d𝑦
d𝑥

= 3𝑥 2

= 3𝑥 2 + 2𝑥

RULES OF DIFFERENTIATION II
5
...
Product

7
...
Inverse function

y = f(u ) where u = g( x )

y = uv

where u and v are
functions of x

u
y=
v

where u and v are
functions of x

y = f( x )

dy dy du
=
dx du dx
dy
dv
du
=u
+v
dx
dx
dx

dy
=
dx
dy
=
dx

v

du
dx

dv

− u dx
v2

1
dx
dy

EXAMPLES FOR RULES 5 − 8:
5
...
Product: given 𝑦 = (𝑥 2 + 1)(𝑥 3 + 𝑥 2 )
Create 2 new variables:𝑢 = 𝑥 2 + 1 and 𝑣 = 𝑥 3 + 𝑥 2
...
Quotient: given 𝑦 =

𝑥 2 +1
𝑥 3 +𝑥 2

Create 2 new variables: 𝑢 = 𝑥 2 + 1 and 𝑣 = 𝑥 3 + 𝑥 2
dv
d𝑢
2
So:
= 3𝑥 + 2𝑥 and
= 2x
d𝑥

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