Calculus

A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks d miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is given by
S = \frac{c}{x^2} + \frac{k}{(d-x)^2}
where c and k are positive constants which depend on the quantity of smoke each stack is emitting. If k = 9 c, find the point on the line joining the stacks where the concentration of the deposit is a minimum.

I know this question has to do with optimizing but I am having a hard time taking the derivative of the equation

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  1. You probably expected this to come out as

    S = c/x^2 + k/(d-x)^2
    = c x^-2 + k(d-x)^-2
    dS/dx = -2cx^-3 - 2k(d-x)^-3
    = 0 for a min

    c/x^3 = k/(d-x)^3
    but k = 9c
    c/x^3 = 9c/(d-x)^3
    1/x^3 = 9/(d-x)^3
    (d-x)^3 = 9x^3
    d-x = (9^(1/3))x
    d = x( 9^(1/3) - 1)

    x = d/(9^(1/3) - 1)

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