Accelerated Calculus
Name______________________
The Derivative of Inverse Functions
Oct. 5/6
Derivative of ln(x)
Let’s consider the function, f ( x)  ln x .
Given the graph of f ( x)  ln x . Sketch its derivative on the axes to the right.
y
y








x








x













Let’s make a conjecture of the derivative of y  ln x :





d
 ln x  
dx
Now, for a “clever” approach to finding the derivative of something we don’t know….but where we DO
know the derivative of its inverse…
Let’s rewrite y  ln x as an equivalent expression we do recognize, with NO ln x : ________________
dy
Now, take the derivative of both sides and see what happens…we want
remember!
dx
Does your answer seem to meet the characteristics of your conjecture from above?
Let’s try this another way, recalling that f  f 1  x    x :
Accelerated Calculus
Derivative of sin-1 (x)
Consider the following: ` y  sin 1  x 
We are trying to find
dy
d
  sin 1  x  
dx dx
Well, as we did with ln(x), we don’t know the derivative of sin-1(x), BUT we do know the derivative of
sin(x) – here is the inverse relationship and its purpose surfacing again!
Let’s rewrite y  sin 1  x  as an equivalent expression we recognize, with NO sin 1  x  : ___________
Now, take the derivative of both sides and see what happens… remember, again we want
dy
!
dx
Does your answer seem to meet the characteristics of your conjecture from above?
Let’s try this another way (as we did with y  ln x ), recalling that f  f 1  x    x :
Q: Suppose we wish to find the derivative of y  sin 1 ( x) at x 
derivative using the function’s inverse.
d
sin 1  x  

dx
x

3
2
3
2
. Determine an expression for its
Accelerated Calculus
Using a similar argument as the one above, one can demonstrate the validity of the following derivatives
of inverse trig functions:


d
1
cos1 x 
dx
1  x2


d
1
tan 1 x 
dx
1  x2
The room below is for you to play around and derive these on your own time!
Derivative of Any Inverse Function:
To summarize, the derivative of an inverse function is given by:
[ f 1 ( x)] 
1
f ( f 1 ( x))
Ex.
Given that f and g are differentiable everywhere, g is the inverse of f, and that f (3)  4 , f (3)  6 and
f (4)  7 .
Find: g (4)
HW: Pg. 141/# 3—30 by 3s, 39, 41, 43, 44, 55, 60