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Homework Help: Finding f'(x) using definition of derivative

  1. May 28, 2009 #1
    f(x) = x2 cos(1/x)

    i know how to take the derivative using product and chain rule, but i need to find the derivative using the definition of the derivative. so far i did:

    lim [(x+h)2 cos(1/x + h) - x2 cos(1/x)] / h
    h ~> 0

    [x2cos(1/x+h) + 2xhcos(1/x+h) + h2cos(1/x+h) - x2 cos(1/x)] / h

    then i took [x2cos(1/x+h)- x2 cos(1/x)] and i factored out the x2

    x2 [cos(1/x+h) - cos(1/x)]

    i used the sum to product formula from trigonometry and i got x2 [-2sin(2x+h / 2x2+2xh)sin(-h / (2x+h / 2x2+2xh)]

    but from there i'm stuck. i have no idea how to simplify that expression in order to get the h on the bottom of the the entire fraction to cancel out so i can substitute 0 for h. please help.
  2. jcsd
  3. May 28, 2009 #2


    User Avatar

    Instead of using the product formula, it may help to note that we takes the limit of h ->0, so [tex]h / x[/tex] is small. This suggests simplification of the argument [tex]\frac{1}{(x+h)} = (x+h)^{-1}[/tex] through binomial expansion.
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