Tuning the Spread of a Bell Curve

A new thermometer is checked five times against a reference. Its errors, in tenths of a degree, are

$$x = 2,\ 9,\ 5,\ 8,\ 6$$

The errors are modelled as coming from a bell curve with centre $\mu$ and spread $\sigma > 0$. How well a choice of $(\mu, \sigma)$ explains the data is scored by its log-likelihood, a function of two variables where bigger is better:

$$\ell(\mu, \sigma) = \sum_{i=1}^{5} \left[\, -\ln \sigma \;-\; \tfrac{1}{2}\ln(2\pi) \;-\; \frac{(x_i - \mu)^2}{2\sigma^2} \,\right]$$

The fit is tuned one dial at a time.

1. Holding $\sigma$ frozen, find the value of $\mu$ at which $\dfrac{\partial \ell}{\partial \mu} = 0$.
2. Keep $\mu$ at that value, and set $\sigma = 3$.

What is $\dfrac{\partial \ell}{\partial \sigma}$ at that point?

Round your answer to 3 decimal places.