Question

(a) Find the differential dy.

Answer 1

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.

$y=\cos x\implies\boxed{\mathrm dy=-\sin x\,\mathrm dx}$

Aside: Compare this to what happens when you differentiate both sides with respect to some other independent parameter, say $t$:

$\dfrac{\mathrm dy}{\mathrm dt}=-\sin x\dfrac{\mathrm dx}{\mathrm dt}$

b. This is just a matter of plugging in $x=\dfrac\pi3$ and $\mathrm dx=0.1$.

$\boxed{\mathrm dy\approx-0.087}$