Derivation of a Derivative

Posted on 2008-11-07
Last Modified: 2012-05-05
Hi, mathe experts...

I am looking at a calculus intoduction and I am not sure how something was derived, I am pretty good at re-arranging normal formulas but a bit stuck here.

How do we get from:

P = C/(1+r) + C/(1+r)^2 + C/(1+r)^3 + ... + C/(1+r)^9 + C/(1+r)^9 + par/(1+r)^10


dP/dr = -C/(1+r)^2 + -2C/(1+r)^3 + ... -10C/(1+r)^11 + -10 par/(1+r)^11

I know thas the diff of x^2 = 2x
Question by:sublimation
    LVL 53

    Accepted Solution

    Since P is a sum, you can differentiate every term separately. They are all of the form :

            C / (1 + r)^n

    This is the same as :

            C * (1 + r)^(-n)

    The derivative of that is :

            C * (-n) * (1 + r)^(-n - 1) * d(1+r)/dr = -C * n / (1 + r)^(n + 1)

    So, for the second term (n = 2) :


    we get the derivative :

            -2C / (1 + r)^3
    LVL 4

    Author Closing Comment

    That is plush!
    LVL 53

    Expert Comment

    btw :

    >> I know thas the diff of x^2 = 2x

    This applies the exact same rule :

            d (x^n) / dx = n * x^(n - 1)

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