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Calculus/Derivative problem

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Question
Hey there,

I'm having a hard time figuring out which rules to use on this function to take the derivative. (i.e. quotient rule, product rule, etc.) I was wondering if you could help me dissect the function to interpret how to take the derivative.

f(x) = 3 - [(8 + 3x)]^2/3

Thanks you very much.

Nick

Answer
If g(x) = 8 + 3x, we have f(x) = 3 - [g(x)]^(2/3) - right?

Now if h(a) = a^(2/3), h'(a) = (2/3)a^(-1/3). Note that z^[(2/3)-1] = z^(-1/3) =1/z^(1/3).

That makes f'(x) = {-(2/3)/[g(x)^(1/3)]}g'(x), and it can be seen that g'(x) = 3.

Since a 3 in the g'(x) cancels out the /3 of the fraction, we get f'(x) = -2/[g(x)^(1/3)].

Put g(x) back in and get f'(x) = -2/[(3x)^(1/3)].

Calculus

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