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Math Help - Find the derivative. The chain rule.

  1. #1
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    Find the derivative. The chain rule.

    I am being asked to find the derivative by what I think is the chain rule of this equation:

    (x^2+2)^2

    Let u be:

    u=(x^2+2)^2

    This is the part I am having trouble understanding. The formula for the derivative of u^2 is given as:

    \frac{d}{dx}u^2=\frac{du}{dx}\frac{d}{du}u^2=(2x)(  2u)=4x(x^2+2)

    Is that the chain rule? How was (2x)(2u) calculated?

    Thanks in advance...
    Last edited by sepoto; November 2nd 2013 at 04:26 PM.
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  2. #2
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    Re: Find the derivative. The chain rule.

    Quote Originally Posted by sepoto View Post
    I am being asked to find the derivative by what I think is the chain rule of this equation:
    (x+2)^2.
    Why do you complicate things?
    \frac{d}{{dx}}{\left( {x + 2} \right)^2} = 2\left( {x + 2} \right)
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  3. #3
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    Re: Find the derivative. The chain rule.

    If you want to use substitution, let u = x^2+2 and then (x^2+2)^2 = u^2 and \dfrac{du}{dx} = 2x. So,

    \dfrac{d}{dx}(x^2+2)^2 = \dfrac{d}{dx}(u^2).

    Then we apply the Chain Rule:

    \dfrac{d}{dx}(u^2) = \dfrac{d}{du}(u^2)\dfrac{du}{dx}

    Notice that on the RHS, you are now differentiating u^2 with respect to u (which is what you want).

    \dfrac{d}{du}(u^2)\dfrac{du}{dx} = (2u)(2x)

    Now, we plug back in for u from our substitution:

    (2u)(2x) = 2(x^2+2)(2x) = 4x(x^2+2)

    So, \dfrac{d}{dx}(x^2+2)^2 = 4x(x^2+2)
    Thanks from sepoto
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  4. #4
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    Re: Find the derivative. The chain rule.

    Quote Originally Posted by sepoto View Post
    I am being asked to find the derivative by what I think is the chain rule of this equation:

    (x^2+2)^2

    Let u be:

    u=(x^2+2)^2
    No. u= x^2+ 2 but that may be a typo.

    This is the part I am having trouble understanding. The formula for the derivative of u^2 is given as:

    \frac{d}{dx}u^2=\frac{du}{dx}\frac{d}{du}u^2=(2x)(  2u)=4x(x^2+2)

    Is that the chain rule? How was (2x)(2u) calculated?

    Thanks in advance...
    Yes, that is the chain rule: if u is a function of f, then \dfrac{df(u(x))}{dx}= \dfrac{df}{du}\dfrac{du}{dx}.
    Here, f(u)= u^2 so \dfrac{df}{du}= 2u and \dfrac{du}{dx}= 2x.
    Then \dfrac{df}{dx}= (2u)(2x)= (2(x^2+ 2))(2x)= 4x(x^2+ 2).
    Thanks from sepoto
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