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calculus

I have been asked to solve this two different ways. The first way is to use the chain rule and then simplify (which I have already done properly), and the second way is to simplify and then differentiate (not necessarily with the chain rule). I am stuck on how to simplify this question and then how to differentiate it.

Square Root(9x^2) ??

Also I am stuck on how to use the chain rule to differentiate and then simplify this question, as well as the second way where I have to simplify and then differentiate it:

(8x^3)^2/3

I would appreciate if someone would be able to explain these two questions to me, thank you.

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Nicole

  1. y = [ 9 x^2 ] ^.5 ??

    let z = 9 x^2 then dz/dx = 18 x

    then we have y = z^.5

    dy/dz = .5 z^-.5 = .5/3x

    but chain rule
    dy/dx = dy/dz*dz/dx (.5 /3x)(18 x)
    = 9/3 = 3 at last

    simplify first
    y = (9 x^2)^.5 = 3x
    dy/dx = 3 the end

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    Damon

  2. (8x^3)^2/3

    I am not going to struggle through that with the chain rule. Go about it the same way as I did upstairs, z = 8 x^3

    simplify method:
    remember (8x^3)^(1/3) = 2x
    (2x)^2 = 4 x^2
    y = 4x^2
    dy/dx = 8 x the end

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    Damon

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