Posted by: Rebekah
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Okay, we have seen that the derivative can be defined as follows:
Please note that the only difference here is that I am using "dx" as a small increase rather than "da".
Also, if we attempt to find a pattern for such derivatives (e.g x², x³), we find the following (please see Table 1.1):
In other words, it appears that, if a function is equal to , then its derivative must be equal to .
But how can we prove this to be true?
Well, using the definition of a derivative above, we are lead to conclude that the derivative must be equal to:
Now all there is left to do is solve!
Before solving, please note the expression c( x + dx )ⁿ : there is no doubt this will require the binomial theorem.
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Okay, now we can start solving!
Now let's multiply through by the c located outside the brackets:
Notice that I was able to place the -cxⁿ inside the bracket because it too was being multiplied by . Consequently, the cxⁿ in the front cancels with the -cxⁿ at the end.
Hey look! Each term inside the brackets has a "dx", so we can factor the "dx" out!
Which is the same as...
It becomes clear that, as x becomes exceedingly small (so much so that it approximates zero), all other terms, with the exception of , also approximate zero.
Thus, we find that the derivative of any function of the form cxⁿ is indeed
Woohoo!!! This means our assumptions were correct!!