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Math Help - Please help with Composite Rule

  1. #1
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    Please help with Composite Rule

    Part2.doc

    Can someone please help as I have tried this and have gone without sleep for 2 nights.

    Thank you.
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  2. #2
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    The composite rule is also known as the chain rule. Put simply, the derivative of the inside times the derivative of the outside.

    \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}

    Where u=\frac{2}{3}x-\frac{1}{6}x^{2}

    The inside, u, is: \frac{4x-x^{2}}{6}=\frac{2}{3}x-\frac{1}{6}x^{2}

    Find the derivative of this and get \frac{du}{dx}=\frac{2}{3}-\frac{1}{3}x

    \frac{dy}{du}=e^{u}. And we know the derivative of e^{u}=e^{u}, so we have:

    \frac{dy}{dx}=\left(e^{\frac{2}{3}x-\frac{1}{6}x^{2}}\right)\left(\underbrace{\frac{2}  {3}-\frac{1}{3}x}_{\text{du/dx}}\right)
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  3. #3
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    In case that was too complicated for you, the following example might be easier. The principle is the same - the derivative of a function is the derivative of the outside times the derivative of the inside.

    Suppose you want to take the derivative of \sin(x^2+3)

    Then your answer is D(\sin(x^2+3)) * D(x^2+3) = \cos(x^2+3) * 2x
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  4. #4
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    composite rule

    can you please show me an example with the first question in my attachment as i just can not get my head around it.

    Do I keep the e in the fuction.

    please help!!!
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  5. #5
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    Quote Originally Posted by omkara View Post
    Do I keep the e in the fuction.
    Basically YES: \frac{{d\left[ {e^{f(x)} } \right]}}{{dx}} = f'(x)e^{f(x)} .
    Examples:
    \frac{{d\left[ {e^{x^3 } } \right]}}{{dx}} = \left[ {3x^2 } \right]e^{x^3 } .

    \frac{{d\left[ {e^{\sin (x)} } \right]}}{{dx}} = \left[ {\cos (x)} \right]e^{\sin (x)}

    \frac{{d\left[ {e^{\tan ^2 (x)} } \right]}}{{dx}} = \left[ {2\tan (x)\sec ^2 (x)} \right]e^{\tan ^2 (x)}
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