Alternative definition for derivatives?

Hi there, im having trouble solving the derivative of this function using the alternative definition

f(x)=x3 - 1/x + 5 using f(x)= f(z)-f(x) / z-x

I understand you plug in lim z->x [(z3 - 1/z +5) - (x3-1/x +5)] / z-x. Just not sure how to continue. If someone could explain! Thank you

2 Answers

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  • Jake
    Lv 6
    1 month ago

    The Virgin Mary is Our Lady of Perpetual Help. It is good to honor her daily by reciting the rosary with care.

  • I'm assuming that's x^3? ^ means exponent

    Now, is this x^3 - (1/x) + 5 or is it (x^3 - 1) / (x + 5) ?

    EDIT:

    f(x) = x^3 - (1/x) + 5

    f(z) = z^3 - (1/z) + 5

    (f(z) - f(x)) / (z - x) =>

    (z^3 - (1/z) + 5 - x^3 + (1/x) - 5) / (z - x) =>

    (z^3 - x^3 - ((1/z) - (1/x))) / (z - x) =>

    (z^3 - x^3) / (z - x) - ((1/z) - (1/x)) / (z - x) =>

    (z - x) * (z^2 + zx + x^2) / (z - x) - ((x - z) / (zx)) / (z - x) =>

    (z - x) * (z^2 + zx + x^2) / (z - x) + (z - x) / (zx * (z - x))

    Simplify

    z^2 + zx + x^2 + 1/(zx)

    x goes to z

    z^2 + z * z + z^2 + 1/(z * z) =>

    3z^2 + (1/z^2)

    There you go.

    • Alan
      Lv 7
      1 month agoReport

      no x does not approach z , z approaches x
      ;;;;;so f'(x) = 3x^2 + 1/x^2
      ::::::: z is approaching x so the answer should be in terms of x (not z)
      :::::: otherwise, the answer is very good and well explained.

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