9 July 2020 LO To differentiate composite functions Review of differentiation So far we have used differentiation to find the gradients of functions made up of a sum of multiples of powers of ID: 931863
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Slide1
Differentiation, The chain rule
9 July 2020
LO: To differentiate
composite functions.
Slide2Review of differentiation
So far, we have used differentiation to find the gradients of functions made up of a sum of multiples of powers of
x. We found that:and when
xn
is preceded by a constant multiplier
k
we have:
Also:
Slide3The chain rule
The chain rule is used to differentiate
composite functions.For instance, suppose we want to differentiate
y = (2x
+ 1)
3
with respect to
x.
One way to do this is to expand
(2x + 1)3
and differentiate it term by term.Expanding using the binomial theorem:
Differentiating with respect to
x
:
Slide4The chain rule
Another approach is to use the substitution u
= 2x + 1 so that we can write
y = (2x + 1)
3
as
y =
u3.
The chain rule states that:
If
y
=
f(u),
u = g(x), then
y =
f(g(
x)), so then
So if
y
=
u
3
where
u
=
2
x
+ 1,
Using the chain rule:
Slide5The chain rule
The derivative of a composite function is the derivative of the outside function with respect to the inside function (inside function remains the same), multiplied by the derivative of the inside function with respect to
xThe chain rule can also be written as:
If
f
(
x
)
= u
(
v(x)), then
f’(
x) = u
’(v(
x)) . v
’(x
)
Slide6The chain rule
Using the chain rule,
Use the chain rule to differentiate
y =
with respect to
x
.
Let
where
u
=
3
x
2
– 5
y =
y =
⇒
Slide7The chain rule
Using the chain rule:
Let
Find
given that
.
where
u
= 7
–
x
3
Slide8The chain rule using function notation
With practice some of the steps in the chain rule can be done mentally.
Suppose we have a composite function
y = g(
f
(
x
))
If we let
y = g(u
) where u = f(
x)
then
and
Using the chain rule:
But
u
=
f
(
x
)
so
If
y
=
g
(
f
(
x
))
then
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