TaylorSeriesDerivative

The Taylor series of a function $f(x)$ centered at a point $a$ is given by:

$$ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \cdots $$

To find the derivative of the Taylor series, simply differentiate each term with respect to $x$:

$$ f'(x) = f'(a) + f''(a)(x-a) + \frac{f'''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n+1)}(a)}{n!}(x-a)^{n-1} + \cdots $$

This is the Taylor series of the derivative $f'(x)$ centered at $a$. Note that the derivative of the Taylor series is equal to the Taylor series of the derivative.