Taylor series operator commutes with differentiation
Suppose if a function defined on a subset of the reals that is infinitely differentiable at a point in its domain. Then, the derivative is also defined and infinitely differentiable at , and the Taylor series for is the derivative (in the sense of derivative of power series) of the Taylor series for .
- Differentiation theorem for power series
- Taylor series operator is linear
- Taylor series operator is multiplicative
Given: is a function defined on a subset of the reals and it is infinitely differentiable at a point in its domain.
To prove: The derivative is defined and infinitely differentiable at and the Taylor series for is the derivative of the Taylor series for .
Proof: We have the following relation between the derivatives of and :
In particular, this means that, since is infinitely differentiable at , so is , with the derivatives given by the above relationship. Thus, it makes sense to take the Taylor series of .
Derivative of the Taylor series
We note that:
Let's try differentiating this. Differentiation of power series is term-wise, so we compute the derivative for each term, i.e., we try to calculate:
The expression is constant, so we use the differentiation rule for power functions on (with the chain rule) and obtain:
For , this becomes 0. For , set (note that now) and this becomes:
Now adding together all the terms, we get:
Since is a dummy variable, we can use the dummy variable instead, and get:
Taylor series of the derivative
We note that:
By , we can rewrite on the right side as , and get:
It is now easy to verify that we have precisely the same expressions for the derivative of the Taylor series of and the Taylor series of .