AP Calculus AB
3-8 Derivatives of Inverse Functions, Day 1

Name_________________________________
I can calculate the numeric derivative of an inverse function
I.
Let’s review:
1.
Let f  x   7 x  1 and g  x  
1a.
Find f g  3
1c.
Find f g  x 
x 1
7
  1 
  2 


1b.
Find f  g    


1d.
How are the functions f  x  and g  x 
related?
2.
Let k  t   2t  5 . Find k 1  t  .
3.
Let h and m be inverses of each other. Find h(m(
)
Let f  x  be some function that is one to one over its domain. Then some function
Theorem:
g  x  is the inverse of f  x  , written f 1  x  iff f  g  x    g  f  x    x
II.
Let f  x   x3
a.
Find f 1  x 
b.
Both f and f 1 are graphed to the right. Sketch the line
y  x . Why would we add this line to the graph?
c.
Find
df
.
dx
g.
e.
f.
df
dx
df
dx
df
dx

h.
x 1

i.
x2

x 3
df 1
.
dx
df 1

dx
df

dx
d.
Find
j.
df 1
dx

x 1
df 1
dx
x 8
df 1
dx
x  27


Conclusion:
* See geometric “proof” on SMART Board. See explanation in text book on page 165.
Theorem:
df
is never zero on I, then f
dx
has an inverse and f 1 is differentiable at every point of the interval f  I 
If f is differentiable at every point of an interval I and
Example 1
Let f  x   x5  3x  2 , and let f 1 denote the inverse of f . Given that 1, 2  is on the graph of f,
df 1
find
dx
x2
Example 2
If f and g are inverses, then we know that f g  x   x .

Use the chain rule to find



d
f  g  x    x . Solve for g '  x  . Explain how your answer makes
dx
sense with the conclusion from the previous page.
Practice 1
Let f  x   x5  2 x3  x  1
(a) Find f 1 and f ' 1
(b) Find f 1  3 and
d 1
f  3
dx
Practice 2
Differentiable functions f and g have values as shown in the table.
a.
b.
c.