a. Practically speaking, you compute the differential in much the same way you compute a derivative via implicit differentiation, but you omit the variable with respect to which you are differentiating.
[tex]y=\cos x\implies\boxed{\mathrm dy=-\sin x\,\mathrm dx}[/tex]
Aside: Compare this to what happens when you differentiate both sides with respect to some other independent parameter, say [tex]t[/tex]:
[tex]\dfrac{\mathrm dy}{\mathrm dt}=-\sin x\dfrac{\mathrm dx}{\mathrm dt}[/tex]
b. This is just a matter of plugging in [tex]x=\dfrac\pi3[/tex] and [tex]\mathrm dx=0.1[/tex].
[tex]\boxed{\mathrm dy\approx-0.087}[/tex]
