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Can you explain the quotient rule and the power rule and how to tell the difference between
them in example. I’m confused how you know the difference and what they mean by the power rule.
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It seems to me that you want to ask something different,
but your question drowned in words.
A reasonable question to ask is THIS:
Let's consider a rational function y(x) = , where n is a positive integer number.
We can consider it as a power function y = , too.
If we want to find the derivative y'(x), we can do it using the power rule
or using the quotient rule.
Is there the difference and how to know, which rule to use ?
The answer is that both these rules lead to the same result in this case,
so one can use any of these two rules.
Indeed, if we use the power rule, then
y'(x) = ' = = . (1)
If we use the quotient rule, then the calculations are as follows
(derivative of the numerator)*denominator - numerator*(derivative of denominator)
y' = ----------------------------------------------------------------------------------- =
denominator^2
0*denominator - 1*n*x^(n-1) (-n)
= ------------------------------ = --------- (2)
x^(2n) x^(n+1)
(1) and (2) are the same, so the answer is the same:
you may use any of these two rules/techniques, the result is the same.