Power rule from limit definition (positive integer exponents)
We will attempt to derive the power rule for derivatives from the limit definition of the derivative. For the purposes of this problem we will limit ourselves to positive integer exponents.
Find the derivative of for positive integers using the limit definition of a derivative. Hint: It may be useful to recall the binomial theorem.
We begin by writing the limit definition of a derivative:
In our case so this becomes
In order to solve this limit is will be necessary to expand the term. To do this, we will use the binomial theorem.
The binomial theorem is given by the following equation:
We do not need to know these coefficients yet, so for now represent them as Thus we have
Finally, we plug this into the limit and solve. Plugging this equation into the limit, we get
We can now cancel out the two terms in the numerator and then divide out an from each of the remaining terms. We are left with
We can now substitute in and note that only the first term remains. So
Recall from above that but we can reduce this as follows:
Hence, our final answer is
which is exactly the power rule.
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