MAT135H1 Lecture Notes - Lecture 6: Function Composition

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Published on 22 Sep 2020
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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
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