Derivative of $x^n$ where $n$ is a positive integer.

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Lesson Parent: 
Differentiation IV: Differentiation Formulas Everyone Must Know
Problem: 

Derivative of $x^n$ where $n$ is a positive integer.

Answer: 

$nx^{n-1}$

SAMPLE SOLUTION

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Solution: 

Recall that for $n=0$, $n=1$, and $n=2$ we have found that

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The proof is by induction where we assume that the result holds for $n=j$ and show that the result holds for $n=j+1$. So now

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Now by the product rule we know that

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so that

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Therefore,

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By the way we have also proven that this formula holds for negative integers.