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Prove that \[ \frac{d}{dx}x^n=nx^{n-1} \]using the Binomial Theorem where $n$ is a positive integer.

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Lesson Parent:

Problem:

Prove that
\[
\frac{d}{dx}x^n=nx^{n-1}
\]using the Binomial Theorem where $n$ is a positive integer.

Solution:

Our function limit definition says that

To solve this let's use the Binomial Theorem and show that

We want to notice that

Now there are three things to note here. First, <Sign in to see all the formulas>. Second, <Sign in to see all the formulas>. Third, because $x$ is a constant in this problem,

is a constant we will call $C_j$. This gives us

Now let's plug it back in.

Next, let's note that

and

and

The proof is now complete.