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Math Help - Differentiation Rules

  1. #1
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    Differentiation Rules

    Hi, I do not understand the rule 5. Is it possible to include an example for me to understand better?

    Thanks.
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  2. #2
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    Re: Differentiation Rules

    That's the chain rule. It's easier to write it like this: \displaystyle \begin{align*} \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \end{align*}. Basically it means that if you have a composition of functions, you need to isolate the "inner" function (u) and work out its derivative, then write y as a function of u and evaluate its derivative. After converting it back to a function of x, multiply the two derivatives together to get the total derivative.
    Thanks from MarkFL
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  3. #3
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    Re: Differentiation Rules

    Here are a couple of examples:

    a) y=\sin(x^3)

    \frac{dy}{dx}=\cos(x^3)\frac{d}{dx}(x^3)=3x^2\cos(  x^3)

    b) y=e^{\tan(2x)}

    \frac{dy}{dx}=e^{\tan(2x)}\frac{d}{dx}(\tan(2x))=e  ^{\tan(2x)}\sec^2(2x)\frac{d}{dx}(2x)=2\sec^2(2x)e  ^{\tan(2x)}
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  4. #4
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    Re: Differentiation Rules

    Quote Originally Posted by MarkFL2 View Post
    Here are a couple of examples:

    a) y=\sin(x^3)

    \frac{dy}{dx}=\cos(x^3)\frac{d}{dx}(x^3)=3x^2\cos(  x^3)

    b) y=e^{\tan(2x)}

    \frac{dy}{dx}=e^{\tan(2x)}\frac{d}{dx}(\tan(2x))=e  ^{\tan(2x)}\sec^2(2x)\frac{d}{dx}(2x)=2\sec^2(2x)e  ^{\tan(2x)}
    Or to use the easier-to-remember form of the chain rule that I posted...

    1. \displaystyle \begin{align*} y = \sin{\left( x^3 \right)}  \end{align*}

    Let \displaystyle \begin{align*} u = x^3 \end{align*} which gives \displaystyle \begin{align*} y = \sin{u} \end{align*}.

    Then \displaystyle \begin{align*} \frac{du}{dx} = 3x^2 \end{align*} and \displaystyle \begin{align*} \frac{dy}{du} = \cos{u} = \cos{\left(x^3 \right)} \end{align*}.

    So \displaystyle \begin{align*} \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3x^2\cos{\left(x^3 \right)} \end{align*}


    See if you can use a similar process with the second example. You will need to use the Chain Rule TWICE.
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  5. #5
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    Re: Differentiation Rules

    Okay. I got it. Thank you Prove It and MarkFL2 for your clear examples and explanation!!
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