Confused in the red areas. This section builds up to defining the exponential as
ID: 3347743 • Letter: C
Question
Confused in the red areas.
This section builds up to defining the exponential as the inverse function of the logarithm. Suppose f(x) = 3x + 4. Then the inverse of f is given by f-1(x) = Moreover, f'(x) = and (f-1)' (x) = In general, if f'(x) = A, then (f-1)'(f(x)) = (Note: Your answer must be in terms of A.) Now let f(x) = x + x5. Then f(1) = f'(1) = f-1(2) = , and (f-1)' (2) = Notice that you can obtain the last two results without knowing a general expression for the inverse function of f. Indeed, if you feel enterprising try to come up with such an expression. Let me know what you find.Explanation / Answer
f inverse x = (x-4)/3
derivative of (f^-1 (x)) = d/dx ( x-4)/3 = 1/3
(f^-1(x)) ' * f(x) = (1/3) * (3x+4)
f inverse x = (x-4)/3
f inverse 2 = (2-4)/3 = -2/3
derivative of (f^-1 (x)) = d/dx ( x-4)/3 = 1/3
so (f^-1(2) ) ' = 1/3