Given f=e^x+x+15 what is (f^-1)'(1).

Since the requestof the problem  is to calculate the
derivative of the inverse of f(x), we'll conclude that the given function is
bijective. 

According to the rule, a bijective function is
invertible.

We know that the product between the derivative
of a function and the derivative of the inverse of the function is
-1.

So, we'll write mathematically the
statement:

[f(x)]'*[(f(x))^-1]' =
1

We infer that (f^-1)' =
1/[f(x)]'

We'll differentiate the function
f(x):

f'(x) = (e^x + x +
15)'

f'(x) = e^x +
1

So,

(f^-1)' (x) = 1 / (e^x +
1)

Now, we'll calculate (f^-1)'
(1):

(f^-1)' (1) = 1 / (e^1 +
1)

(f^-1)' (1) =
1/(e+1)

(f^-1)'
(1) = 0.264550...