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July 29th, 2011, 12:03 AM   #1
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Variables and unknown constants?

Hi,



Through a maths course I have been learning about differentiating inverse trig functions. I just learnt the rule:





I understand how to use the rule and how to prove the rule.




I was recently shown a question and to solve it they did

And then continued making it more elegant






But what I dont understand is why were they able to use the same rule and just sub it into the end?




I mean if you got

You couldnt just say that






Because that would be wrong because you need to use chain rule.




So why can a be pretty much anything but x must remain as x for you to be able to use the rule?



I tried to ask my teacher and he said it is because a is a constant. That confused me even more because I am not sure what the difference is between a variable and an unknown constant. Seeing as they both can take on any value.



Do constants have some sort of property where you can replace them for anything in a rule and the rule still works while variables cant be replaced without creating a new rule?



Sorry if this is worded badly I am having a bit of trouble trying to explain why I am confused. I would really appreciate any help.



Thank you
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July 29th, 2011, 12:08 AM   #2
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Re: Variables and unknown constants?

Since you are differentiating with respect to x, any expression or term not containing x, or a function of x, is treated as a constant during the differentiation.
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July 29th, 2011, 12:21 AM   #3
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Re: Variables and unknown constants?

Quote:
Originally Posted by MarkFL
Since you are differentiating with respect to x, any expression or term not containing x, or a function of x, is treated as a constant during the differentiation.
Thanks for the quick reply. Ok so I understand that "a" is a constant but why is it that you can replace constants for anything (in this case ) and just sub the new value into making it . But you can't do the same for variables and replace with and sub that into making it . Is there some property of constants that allows you to do that?
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July 29th, 2011, 05:09 AM   #4
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Re: Variables and unknown constants?

That's just like saying then . You can only do such things with constants, because in your case is also a constant.
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July 30th, 2011, 11:53 AM   #5
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Re: Variables and unknown constants?

Quote:
Originally Posted by delta_salt
Is there some property of constants that allows you to do that?
As [color=#00AA00]MarkFL[/color] pointed out, the value of the constant remains the same no matter what the value of the variable we are differentiating with respect to (in this case, x). The constant's rate of change, with respect to x, is zero. Using a letter in place of a specific number allows us to generalize statements to accommodate any given constant, so rules of differentiation and the like can be written more compactly.

Any operation applied to a constant cannot involve x, otherwise the value of the constant would depend on x and we would have to account for that when we differentiate. Performing an operation on a constant that does not involve x results in another constant, i.e. its value does not depend on x, and its rate of change w.r.t.x. remains zero.
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July 31st, 2011, 02:33 AM   #6
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Re: Variables and unknown constants?

Suppose the problem had been to differentiate , would you have had any problem using the fact that the derivative of is to say that the derivative of is just by replacing the symbol "a" by the number "5"? I would hope not- that is the whole point of algebra! We can use letters to represent numbers and replace them any way we wish. And if a represents "any number", the surely is also a number! We are simply replacing "a", which represents any number, by which also represents a number.
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July 31st, 2011, 06:01 AM   #7
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Re: Variables and unknown constants?

Thanks so much for the replies. I get it now!
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