# 4.3 - MATH 1081  Wednesday, February 16   ...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 1081  Wednesday, February 16    Chapter 4 – Section 3    THE CHAIN RULE    Homework #5 (due 2/21):  Section 4.2 #8, 22, 32, 46, 52                  Section 4.3 #34, 38, 56, 62    Clicker Check­in:  Choose any letter to check in now.    Example:  Suppose that  f ( x )  and  g ( x )  are differentiable    functions at  x = 2  with   f ( 2 ) = 7 ,  f ' ( 2 ) = 3,  g ( 2 ) = 5 , and  g ' ( 2 ) = −4 .      5 f ( x) 1. Find  q ' ( 2 ) when  q ( x ) = .  g ( x ) − 10         2. Find  p ' ( 2 )  when  p ( x ) = x 2 f ( x ) .      Consider the function  f ( x ) = x − 5 .  At this point, to find the  derivative of this function, we would either need to multiply it  out to get the polynomial  f ( x ) = x 6 − 10 x 3 + 25  or write it as a  product  f ( x ) = x 3 − 5 x 3 − 5 .    For this particular function, either of these options is fairly  simple.  However, as soon as we either start to increase the  21 3 power on the outside to  f ( x ) = x − 5  or we make the  function on the inside more complicated like  2 3 2 ⎛ x + 4x − 5⎞ , doing either of the above options  f ( x) = ⎜ 1/ 2 ⎟ ⎝ 9x − 8x ⎠ becomes more and more complicated.  3 2 ( ) ( )( ) ( ) Today, we will look at the chain rule that will help us with  these more complicated functions.  In particular, the chain rule  shows us how to find the derivative of a composite function.    Definition:  Composite function    Let  f  and  g  be functions.  The composite function, or  composition, of  g  and  f  is the function whose values are  given by  g ⎡ f ( x ) ⎤  for all  x  in the domain of  f  for which the  ⎣ ⎦ value  f ( x )  is in the domain of  g .    In other words, a composite function is one built by plugging  one function into another.  Plug an  x  into  f  and take the  y  that  comes out of  f  and plug that into  g .  The value that comes out  of  g  is the result.  Example:  Find the compositions  g ⎡ f ( x ) ⎤  and  f ⎡ g ( x ) ⎤  of the   ⎣ ⎦ ⎣ ⎦  functions.    x       1.   f ( x ) = −8 x + 9  and  g ( x ) = + 4   5              2.   f ( x ) = 9 x 2 − 11x  and  g ( x ) = 2 x + 2               3.   f ( x ) = x 21  and   g ( x ) = x 3 + 4 x 2 − 5   Rules of Differentiation      7.  Chain Rule        If  f ( x ) = u ⎡ v ( x ) ⎤ , then  f ' ( x ) = u ' ⎡ v ( x ) ⎤ ⋅ v ' ( x ) , assuming   ⎣ ⎦ ⎣ ⎦     all relevant derivative exist.      In other words, take the derivative of the outside function,  leaving the inside alone.  Then, take the derivative of the inside  function.  So, we are creating a “chain” working from the  outside in.    Example:  Find the derivative of the function.    2 3       1.   f ( x ) = x − 5         3/ 2 2       2.   g ( t ) = 5t + 9t − 8           x +1       3.   y = 5    3x 2 − 11x + 4 ( ) ( ) ( ) Example:  Consider the table of values of functions and their    derivatives at various points.    x  1  2  3  4  f ( x)  f '( x )  g ( x)  g '( x )  2  ‐6  2  2/7  4  ‐7  3  3/7  1  ‐8  4  4/7  3  ‐9  1  5/7    Find  Dx f ⎡ g ( x ) ⎤ and  Dx g ⎡ f ( x ) ⎤  at  x = 1 and  x = 2 .  ⎣ ⎦ ⎣ ⎦ ( ) ( ) Clicker Check­out:  Choose any letter to check out now.      Tomorrow in recitation:  Workshop #3    Next time – Section 4.4     ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online