I think you can always do that. Differentiating is really no different from any other mathematical function. You don't say that you're not allowed to rearrange before adding things, so why should differentiating be any different?

Differentiating the products and quotients of trig functions tend to create a monster, since the product and quotient rule for differentials tend to produce expressions that are a bit on the long side. This tends to lead to a trigonometric identity nightmare! What I'm wondering is, is it ok to rearrange the function, before you differentiate?

This is the problem I had.

sin(2t)/cos^2 t

If you first replace sin(2t) with 2sin t cos t you have:

2 sin t cos t / cos^2 t

you can cancel out cos t and get

2 sin t/ cos t

But sin t /cos t = tan t so we can rewrite it as

2 tan t

differentiate and we get:

2 sec^2 t

Now thats easy as pi and it was the correct answer to the problem. But would rearranging before differentiating ever produce an ambigous answer?

Of course, the form of the original function must always be taken into acount, we cannot use values of t that would result in division by zero or the square root of negative numbers in the original function, even if those values work fine in the deriviate of the function.

Anyways, the question is, is rearranging before differentiating ever a bad idea?