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Math Help - Proof by induction for power rule in derivatives

  1. #1
    Newbie superphysics's Avatar
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    Proof by induction for power rule in derivatives

    I'm looking for a proof by induction for the power rule of derivatives, i.e. \frac{d}{dx}x^n=nx^{n-1} for n\geq1

    In the proof, I need to use the following facts:

     <br />
x^{n+1}=xx^n<br />
    and
     <br />
g'(x)=x(f'(x))+f(x)<br />

    Can someone help out here?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by superphysics View Post
    I'm looking for a proof by induction for the power rule of derivatives, i.e. \frac{d}{dx}x^n=nx^{n-1} for n\geq1

    In the proof, I need to use the following facts:

     <br />
x^{n+1}=xx^n<br />
    and
     <br />
g'(x)=x(f'(x))+f(x)<br />

    Can someone help out here?
    Base case:

    <br />
\frac{d}{dx}x^1 = \frac{d}{dx}x = 1<br />

    Prove this directly from the definition of the derivative.

    Suppose for some k>0\ k \in \mathbb{N} that

    \frac{d}{dx}x^k=kx^{k-1}

    Then consider:

    \frac{d}{dx}x^{k+1}= \frac{d}{dx}[x \ x^k]=x^k + x\ (kx^{k-1})=(k+1)x^k

    The second from last step uses the product rule the assumption and the base case.

    From there it is routine to complete the proof

    RonL
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