For every natural number n, the derivative of is
using the product rule:
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I have worked out
Is this correct so far?
Using the product rule is useless. n is a constant and its derivative is 0 since the slope of a constant (e.g. y=1) is 0.
If you want to prove that then I fear that you have no choice but to compute
I guess that you can use the intuitive notion of a limit to solve this.
Expand the whole thing, simplify (substract ), divide by and compute the limit (everything goes to 0 except one term).
Let P(n) be the statement we want to prove.
We proceed by induction on , where
Clearly is true, since ....(of course, . if you want x to be anything it wants to be, then take the natural numbers greater than or equal to 1 in your statement)
Assume is true, we show is true.
Now,
basically, you want to apply the product rule, and simplify to get , which would be
i do not understand what you mean. use the product rule to differentiate . we can differentiate each of those terms because is true. so we can validly apply the product rule here
although, i am thinking of starting with P(1) instead of P(0). what is the definition of natural numbers that you are using. including zero has some annoying cases that we may have to deal with explicitly.