Show that

where

Since we do not have the binomial theorem for real powers (we have proved a formula for for integers , but in this case the power we have is a real number ), we use induction to determine the th derivative of the function . First, we compute a few derivatives of ,

So, we conjecture

We have shown this is true for , and if we suppose it is true for a positive integer then we have

Therefore, the formula holds for all positive integers. Now we can compute the Taylor polynomial directly,