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Using the Binomial Theorem and the dfinition of the derivative of a function

  1. Apr 16, 2009 #1
    Using the Binomial Theorem and the definition of the derivate of a function

    f(x) as f'(x)= lim as h tends to 0 ((f(x+h)-f(x))/h)

    Prove that if f(x)=x^n

    then

    f'(x)=nx^(n-1)



    I'm confused as to how to exactly incorporate the nCr "n choose r" into this interpretation of the derivative.

    Any hints or explanations would be greatly appreciated!
     
  2. jcsd
  3. Apr 16, 2009 #2
    Write nCr using its definition:

    nCr = n! / [r! * (n-r)!]
    Notice that nC1 = n! / [1! * (n-1)!] = (n*(n-1)* ... *2*1)/[(n-1)*(n-2)*...*2*1] = n

    Also, the first term in the binomial expansion of (x + h)^n is x^n, so that will disappear when you subtract f(x) = x^n.

    The remaining terms all have a factor of h... interesting...
     
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