How to prove this with mathematical induction?How do you prove the power rule of derivatives [d/dx x^n = nx^(n-1)] using mathematical induction?  I know the concept of how to use mathematical...

How to prove this with mathematical induction?

How do you prove the power rule of derivatives [d/dx x^n = nx^(n-1)] using mathematical induction?  I know the concept of how to use mathematical induction, I just don't really know where to start on this problem specifically.

Asked on by wfc2008

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Prove `d/(dx)x^n=nx^(n-1)` , for `n in NN` using mathematical induction.

(1) Base case: If n=1 then `d/(dx)x^1=1x^0=1` which is true.

(2) Inductive hypothesis: Assume for some n>1that for all `1<=k<=n` , `d/(dx)x^k=kx^(k-1)`

(3) We need to show that for such an n, `d/(dx)x^(n+1)=(n+1)x^n` .

Now `x^(n+1)=x*x^n` . We take the derivative of this, using the product rule:

`d/(dx)[x*x^n]=x*d/(dx)[x^n]+d/(dx)[x]*x^n`

`=x(nx^(n-1))+x^n` by the inductive hypothesis

`=nx^n+x^n`

`=(n+1)x^n` as required. Q.E.D.

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