# MAT135H1 Lecture Notes - Lecture 6: Function Composition

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2.5 The Derivative of Composite Functions

Composite Function:

• Given two functions

()

fx

and

()

gx

, a composite function is defined as:

Example 1: Given ()

fx x

= and

() 5

gx x

=+

, determine

a)

(())

fgx

b)

((4))

fg

The Chain Rule:

• If ‘f’ and ‘g’ are functions, then the derivative of the composite function

() ( ())

hx f gx

=

is

• Leibniz notation:

! if ‘y’ is a function of ‘u’ and

! if ‘u’ is a function of ‘x’ then

• Note: the Power of a Function Rule (Section 2.3) is a special case of the Chain Rule:

Example 2: If 3

21

yu u

=−+

where

2

ux

=, find

dy

dx

at

4

x

=

fglad of gCfCx

Txt

914 45fact fs

fg

3

hx_f gcn g'Cx

ddxt ddf.dz

421

atx 4

ddidnt du254

3u2 2xt44

at xu 4y 445 24

48 2ft

46 I

4237