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Math Help - Find the Derivative

  1. #1
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    Find the Derivative

    Just started derivatives in Math 262 and we are learning the chain rule.
    Then I get this:

    f(x) = 2x + (2x + (2x +1)^3)^3

    Her is what I'm attempting...

    2x + 3(2x + 3(2x + 1)^2)^2

    Is this even close?

    Thanks for any help you might have.
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  2. #2
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    Re: Find the Derivative

    Quote Originally Posted by KonaBear View Post
    Just started derivatives in Math 262 and we are learning the chain rule.
    Then I get this:

    f(x) = 2x + (2x + (2x +1)^3)^3

    Her is what I'm attempting...

    2x + 3(2x + 3(2x + 1)^2)^2

    Is this even close?

    Thanks for any help you might have.
    First of all, the derivative of a sum is equal to the sum of the derivatives. So the derivative of the first 2x is 2. The hard part will be evaluating the derivative of \displaystyle \begin{align*} \left[ 2x + \left( 2x + 1 \right) \right]^3 \end{align*}. It's a composition of functions, so the chain rule will need to be used. I always use Leibnitz notation for the chain rule, since it is easier. In this case you will need to do \displaystyle \begin{align*} \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \end{align*}.

    So first, if we have \displaystyle \begin{align*} y = \left[ 2x + \left( 2x + 1 \right)^3 \right]^3 \end{align*}, then we let \displaystyle \begin{align*} u = 2x + \left( 2x + 1 \right)^3 \implies y = u^3 \end{align*}. Then \displaystyle \begin{align*}\frac{dy}{du} = 3u^2 = 3 \left[ 2x + \left( 2x + 1 \right)^3 \right]^2  \end{align*}.

    As for finding \displaystyle \begin{align*} \frac{du}{dx} \end{align*}, we notice that \displaystyle \begin{align*} u = 2x + \left( 2x + 1 \right)^3 \implies \frac{du}{dx} = 2 + \frac{d}{dx} \left[ \left( 2x + 1 \right)^3 \right] \end{align*}. Can you use the Chain Rule to evaluate this final derivative?
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  3. #3
    MHF Contributor MarkFL's Avatar
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    Re: Find the Derivative

    Post edited to prevent devaluation of above post...
    Last edited by MarkFL; October 3rd 2012 at 09:26 PM.
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  4. #4
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    Re: Find the Derivative

    Ok so the final answer would be

    2 + 3[2x + (2x+1)^3]^2 (2 + 3(2x + 1)^2) ?
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    Re: Find the Derivative

    Quote Originally Posted by KonaBear View Post
    Ok so the final answer would be

    2 + 3[2x + (2x+1)^3]^2 (2 + 3(2x + 1)^2) ?
    Not quite, it'll actually be \displaystyle \begin{align*} 2 + 3\left[ 2x + \left( 2x + 1 \right)^3 \right]^2 \left[ 2 + \mathbf{6}\left( 2x + 1 \right)^2 \right] \end{align*}. Why?
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    Re: Find the Derivative

    I wish I could see it, but I don't. No idea where that 6 comes from.
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  7. #7
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    Re: Find the Derivative

    When you evalute \displaystyle \begin{align*} \frac{d}{dx}\left[ \left(2x+ 1 \right)^3 \right]  \end{align*}, the "inner" function is u = 2x + 1. What's its derivative?

    The "outer" function is \displaystyle \begin{align*} u^3 \end{align*}. What's its derivative?

    What do you get when you multiply them together?
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  8. #8
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    Re: Find the Derivative

    Ohhhhh ok! Thank you so much!
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