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Thread: product rule, taking derivatives

  1. #1
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    product rule, taking derivatives

    2. Find the derivative of $\displaystyle y = 3 sin^{3} (2x^{4} + 1) $

    I used the product rule and this is my working, however I don't end up with the correct answer, can someone please tell me where I am going wrong, thanks.

    $\displaystyle u = 3sin^{3} $

    $\displaystyle v = 2x^{4} +1 $

    $\displaystyle t = sin , y= 3t^{3} $


    $\displaystyle \frac{dt}{dx} = cos $

    $\displaystyle \frac{dy}{dt} = 9t^{2} $

    so now I multiply both together to get the derivative of $\displaystyle 3sin^{3} $

    $\displaystyle 9sin^{2} \times cos $

    $\displaystyle \frac{dy}{dx} = 8x^{3} $

    Now using the product rule and putting this together.


    $\displaystyle 9sin^{2} \times cos \times (2x^{4} +1) + 8x^{3} \times 3sin^{3} $

    the correct answer is.
    $\displaystyle 72x^{3} sin^{2}(2x^{4} +1)cos(2x^{4} +1) $
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  2. #2
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    not the product rule ... the chain rule

    $\displaystyle y = 3[\sin(2x^4+1)]^3$

    $\displaystyle
    y' = 9[\sin(2x^4+1)]^2 \cdot \cos(2x^4+1) \cdot 8x^3
    $

    now clean up the algebra
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  3. #3
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    Quote Originally Posted by skeeter View Post
    not the product rule ... the chain rule

    $\displaystyle y = 3[\sin(2x^4+1)]^3$

    $\displaystyle
    y' = 9[\sin(2x^4+1)]^2 \cdot \cos(2x^4+1) \cdot 8x^3
    $

    now clean up the algebra
    Oh I see.

    But I am just wondering how did you know to use the chain rule instead of the product rule? As this expression is a product of two functions, so that's why I assumed I should use the product rule.

    and I thought the chain rules was supposed to be used for a function of a function. Like $\displaystyle \sqrt{3x-1} $

    So for an expression like this $\displaystyle x^{2} \sqrt{3x-1} $

    would you use the chain rule or product rule?

    thank you for clarifying, greatly appreciated.
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  4. #4
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    Quote Originally Posted by Tweety View Post
    Oh I see.

    But I am just wondering how did you know to use the chain rule instead of the product rule? As this expression is a product of two functions, <<<< No. See below
    so that's why I assumed I should use the product rule.

    and I thought the chain rules was supposed to be used for a function of a function. <<<<< Correct

    ...
    Your function is indeed the function of a function of a function:

    $\displaystyle f_1(x)=2x^4+1$

    $\displaystyle f_2(x)=\sin(x)$

    $\displaystyle f_3(x)=3x^3$

    And your function becomes:

    $\displaystyle f(x)=f_3(f_2(f_1(x)))$

    As Skeeter indicated the powers of trigonometric functions are written at the name of the function:

    $\displaystyle \sin^2(x)=(\sin(x))^2$

    and - very important - mostly this unequality is true:

    $\displaystyle \sin^2(x) \neq \sin((x)^2)$
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