First & Second Derivative of a Summation

Question

Consider the function $f(\mu) = \sum_{i = 1}^{n} (x_i - \mu)^2$, where $x_i = i,\,i=1, 2,\dots, n$.

What is the first and second derivative of $f(\mu)$ ?

Answer 1

$f'(\mu) = -2\sum_{i = 1}^{n} (x_i - \mu)$ and $f''(\mu)=2n$

Answer 2

$$\frac{d}{d \mu} f(\mu) = -2 \sum_{i=1}^n (x_i - \mu) = -2 \sum_{i=1}^n x_i + 2 n \mu $$

$$\frac{d^2}{d \mu^2} f(\mu) = 2 n $$