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Thread: 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:

    $\displaystyle (x^2+2)^2$

    Let u be:

    $\displaystyle u=(x^2+2)^2$

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

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

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

    Thanks in advance...
    Last edited by sepoto; Nov 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:
    $\displaystyle (x+2)^2$.
    Why do you complicate things?
    $\displaystyle \frac{d}{{dx}}{\left( {x + 2} \right)^2} = 2\left( {x + 2} \right)$
    Thanks from sepoto
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  3. #3
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    Re: Find the derivative. The chain rule.

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

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

    Then we apply the Chain Rule:

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

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

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

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

    $\displaystyle (2u)(2x) = 2(x^2+2)(2x) = 4x(x^2+2)$

    So, $\displaystyle \dfrac{d}{dx}(x^2+2)^2 = 4x(x^2+2)$
    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:

    $\displaystyle (x^2+2)^2$

    Let u be:

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

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

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

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

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