(A) Using the notation of h'(x) for the derivative, d/dx [ h(x) ], of a function h(x), recall that the Chain Rule (this "rule" is actually a theorem) states that:

d/dx [ f(g(x)) ] = ____________________   (1)

Suppose now that f(x) and g(x) are inverse functions . It follows that f(g(x)) equals x, the identity function. When f(x) and g(x) have this relationship, you can re-write your answer in the above blank space. Copy your answer to part (A) above in the left blank space below, then use the fact that f and g are inverses to re-write it:

We know that, when when f(g(x))=x, then

_________________ = d/dx [ f(g(x)) ]   =   _______   (2)

Thus, solving for g'(x), we get g'(x)   =   ________.   (3)

(B) Now let's consider the case when f(x) = e^x. In this case, we can rewrite equation (3) while leaving expressions with 'g' in them alone (not plugging anything in for them) but just using that f(x) = e^x to find and substitute for f'(x). This gives us:

  g'(x)   = ________________ (4)

Now using the relationship between f and g, we simplify once more to arrive at:

  g'(x)   = ________________ (5)

(C) Conclusion: We have discovered the derivative of a certain function. The function is ________ and we've proved that its derivative is ________.