Question:Talyor expansion meets the time derivative

Question:Talyor expansion meets the time derivative

Maple

Sorry for disturbing you again. I am haunted by a problem about Taylor expansion in Maple.

For example, when the angle alpha is small, thus we could use taylor(sin(x),x=0,1) to get a approximated function of sin(x) with different order of precision.

But, if the sin(x) meets diff(x(t),t), (x is a time-dependant variable), could we also apply the Tayolr expansion in Maple to obtain a approximation?

For example, the function: sin(x(t))*diff(x(t),t), x(t) is given with a small value, but diff(x(t),t) could not be neglected, because even x(t) is with a small value , diff(x(t),t) could be great. We could conduct the Talyor expansion with hands, and sin(x(t))*diff(x(t),t) becomes x(t)*diff(x(t),t). But I wonder if we could use Maple to realize this procedure. And if the Maple could make it, I would like to know if the Maple could handle the multi-variable Taylor expansion accompanied with the time derivative. For example, the x(t) and y(t) are considered as small value variables, could we simplified the following function sin(x(t))*cos(y(t))*diff(x(t),t)*diff(y(t),t).

Thank you very much in advance for taking a look.

﻿