Are there any functions where differentiating the Taylor series yields an accurate approximation for the differentiated function?

Question

Considering the Taylor series for $e^x$, you can differentiate it and get the same series. ie $$\frac{d}{dx} (1 + x + \frac{x^2}{2!} + ...) = 1 + x + ...$$

This is obviously a property of $e^x$, thus implying that differentiating the Taylor series yields the series for the differentiated function. Is this a valid property, or just a fluke because it's $e^x$? And does this hold true for other functions with Taylor series--can you differentiate other series and still get an accurate representation of the differentiated function? If you can explain the underlying analysis that would be very helpful too.

Thanks :)

Answer 1

If $f(x) = \sum_{n=0}^{\infty}a_nx^n, x\in (-r,r),$ then

$$f'(x) = \sum_{n=1}^{\infty}na_nx^{n-1}, x\in (-r,r).$$

This is a standard result in the theory of Taylor series.