The Dial With a Reciprocal Cost

A model has a single dial $w > 0$ . Its total loss is the sum of two competing pieces:

Piece	Formula	Behaviour as $w$ grows
Complexity penalty	$3w$	climbs steadily
Fit error	$\dfrac{12}{w}$	falls away

$L(w) = 3w + \frac{12}{w}$

Before the optimizer can take a single step it has to ask the loss one question: how steep is it right here? The dial currently sits at $w = 4$ .

A slope worked out for one setting is useless the moment the dial moves, so do it properly. Working from the limit definition

$\frac{dL}{dw} = \lim_{h \to 0} \frac{L(w + h) - L(w)}{h}$

first find the derivative as a function of $w$ , then evaluate that function at $w = 4$ .

Round your answer to two decimal places.