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#1 2005-12-01 12:38:50

mikau
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differantiating products and quotients of trig functions

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?

Last edited by mikau (2005-12-01 12:42:16)


A logarithm is just a misspelled algorithm.

#2 2005-12-01 18:14:07

ryos
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Re: differantiating products and quotients of trig functions

The derivative (as you know) is the slope of the function. If two functions, rearranged algebraically, are truly equivalent, then they should also have the same graph, meaning the same slope, or derivative.

At least, that's what I think, but math can be a strange beast and surprise unexpectedly. Can anyone think of an exception?


El que pega primero pega dos veces.

#3 2005-12-02 04:21:39

mathsyperson
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Re: differantiating products and quotients of trig functions

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?


Why did the vector cross the road?
It wanted to be normal.
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