DERIVATIVES OF INVERSE
FUNCTIONS
5-E
Inverse Functions
(1) Two functions are
inverses if
f g ( x)  g f ( x)  x
(2) The graph of f contains
the point (a,b) if and
only if the graph of the
inverse contains (b,a)
(3) Reflect over the line
y=x
Inverse Functions
At right are the graphs of
a function f(x) and its
inverse f-1(x).
Do you see a relationship
between the slope of the
graph of f at (a,b) and the
slope of the graph of the
inverse at (b,a)?
•
•
To Find Inverses
• Solve the equation for x
• Interchange x and y
• Replace y with
f 1 x 
1) let
f (x) 
x
a) sketch the graph
1) let
f (x) 
x
b) find the inverse
c) sketch the inverse
1) let
f (x) 
x
d) differentiate both f(x) and f-1(x).
1) let
f (x) 
x
e) find the slope of the graph of f(x) at (4, 2) and
the slope of the inverse at (2, 4)
f) What conclusion can you make about these
slopes?
conclusion
Since slope m = dy/dx, it should make sense
that switching x and y (for inverse
functions) should produce reciprocal
slopes for inverse functions
1
2) Find the derivative of the inverse of y  x  1
2
a) Find inverse then differentiate
Derivative of an inverse
1
f
( x) ,
If (a, b) is a point on f, then (b, a) is a point on
1
then ( f )' (b) 
1

df 1
f (b)
dx

Or if g is the inverse of f. If f is differentiable then
1
g ( x) 
f ( g ( x))
Take the reciprocal derivative
Proof
[ f ( f 1 )' ( x)]  x
d
d ( x)
1
[ f ( f )( x)] 
dx
dx


f ' ( f 1 )( x)  ( f 1 )' ( x)
1

1
f ' ( f )( x)
f ' ( f 1 )( x)
( f
1

)' ( x) 
1
1
f ' ( f )( x)
dy
1

dx dx
dy
3) Find the derivative of the inverse
of g ( x)  x3  1 at x  9
4) Find the derivative of the inverse
of f ( x)  x  4 at x  12
2
5) Find the derivative of the inverse
of y  2 x  12 at x  14
3
HOME WORK
Derivatives of Inverse
functions Worksheet