Imagine we are using a simple neuron model, with one sigmoid neuron, which takes one input and has one output $--$ and we are using a mean squared error as the cost function.

The change in cost given the change in weight is $($a$-$y$)*$ sigma' $($z$).$ where x is the input $(1$ or $0),$ a is the activation and y is the predicted value $-$

where z $=$ w $*$ x$+$b$,$ and sigma' is the first derivative of the sigmoid

Assuming that $($a$-$y$)=1$ then:

Select one:

a$.$ If the z value is large $($say greater than $100),$ then dC$/$dw will be large$($i$.$e$.$ much greater than zero$).$

b$.$ If the z value is small $($say less than $1),$ then the dC$/$dw will be large $($ie$.$ much greater than zero$).$

C$.$ If the z value is close to zero, then the dC$/$dw will be small $($closer to zero$).$