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Math Help - Confusing Algebra in a Derivative

  1. #1
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    Confusing Algebra in a Derivative

    Hi, I couldn't find anyone around my dorm that could help me so I thought I would try a forum. Thanks for any help I receive!

    My problem is:


    Find the derivative:
    y = (3x^4 + 1)^4 * (x^3 + 4)

    According to my book the answer is 3x^2(3x^4 + 1)^3 * (19x^4 + 64x + 1)

    For some reason I never can get this solution.

    Here are the 2 steps in the book that really confuse me:
    3x^2(3x^4 + 1)^4 + 48x^3(x^3 + 4)(3x^4 + 1)^3
    to this
    3x^2(3x^4 + 1)^3[3x^4 + 1 + 16x(x^3 + 4)]

    I get to the first step but I do not get the second step. I think I am just overlooking something simple, but I don't see it.

    Thanks!!
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  2. #2
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    y = (3x^4 + 1)^4 * (x^3 + 4)

    To find the derivative, we're going to need to use a chain rule and a product rule.

    y'= (3x^4 +1)^4 * d/dx (x^3 +4) + (x^3+4) * d/dx (3x^4 +1)^4
    = (3x^4 +1)^4 * 3x^2 + (x^3+4) * 4(3x^4 +1)^3 * 12x
    = 3x^2(3x^4 +1)^3 + 48x(x^3+4)(3x^4 +1)^3

    From here, all you need to do is simplify the expression to get the textbook answer.

    Oh, I see, didn't read your question thoroughly enough right away. Um, all you really should have to do is simplify the expression and then factor out a 3x^2(3x^4 + 1)^3 from the expression.
    Last edited by Math Major; March 29th 2009 at 05:23 PM. Reason: Completion
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  3. #3
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    Quote Originally Posted by stunirvana View Post
    Hi, I couldn't find anyone around my dorm that could help me so I thought I would try a forum. Thanks for any help I receive!

    My problem is:


    Find the derivative:
    y = (3x^4 + 1)^4 * (x^3 + 4)

    According to my book the answer is 3x^2(3x^4 + 1)^3 * (19x^4 + 64x + 1)

    For some reason I never can get this solution.

    Here are the 2 steps in the book that really confuse me:
    3x^2(3x^4 + 1)^4 + 48x^3(x^3 + 4)(3x^4 + 1)^3
    to this
    3x^2(3x^4 + 1)^3[3x^4 + 1 + 16x(x^3 + 4)]

    I get to the first step but I do not get the second step. I think I am just overlooking something simple, but I don't see it.

    Thanks!!
    3x^2(3x^4 + 1)^4 + 48x^3(x^3 + 4)(3x^4 + 1)^3<br />

    \textcolor{red}{3x^2(3x^4 + 1)^3} \cdot (3x^4+1) + \textcolor{red}{3x^2} \cdot 16x(x^3 + 4)\textcolor{red}{(3x^4 + 1)^3}

    note what factors the two terms have in common ...

    both have the common factors \textcolor{red}{3x^2} and \textcolor{red}{(3x^4+1)^3} ... they are both factored out from each term ...

    \textcolor{red}{3x^2(3x^4+1)^3}[(3x^4+1) + 16x(x^3+4)]

    any clearer now?
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  4. #4
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    I can get this. I think it should be:
    Quote Originally Posted by Math Major View Post
    y = (3x^4 + 1)^4 * (x^3 + 4)
    y'= (3x^4 +1)^4 * d/dx (x^3 +4) + (x^3+4) * d/dx (3x^4 +1)^4
    = (3x^4 +1)^4 * 3x^2 + (x^3+4) * 4(3x^4 +1)^3 * 12x^3
    = 3x^2(3x^4 +1)^4 + 48x^3(x^3+4)(3x^4 +1)^3
    Correct me if I am wrong above.


    Its the simplification I'm having trouble with.
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  5. #5
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    Quote Originally Posted by skeeter View Post

    any clearer now?
    So I just had to pull a (3x^4+1) out of  (3x^+1)^4 then pull them (I mean the two terms you highlighted) out to the front?

    By the way, the math text thing is awesome!
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  6. #6
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    Yes, you're correct. I just typed it too quickly.
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