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Math Help - Implicit differentiation

  1. #1
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    Implicit differentiation

    Hi

    We are asked to solve the following by Implicit differentiation, though I have never encountered this before. Here is the equation

    (x+y)\sin(xy)=1

    Can anyone please help and explain what they are doing?

    Thanks
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  2. #2
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    Quote Originally Posted by bobred View Post
    Hi

    We are asked to solve the following by Implicit differentiation, though I have never encountered this before. Here is the equation

    (x+y)\sin(xy)=1

    Can anyone please help and explain what they are doing?

    Thanks
    \frac{d}{dx}[(x + y)\sin{(xy)}] = \frac{d}{dx}(1)

    Use the product rule on the LHS

    (x + y)\,\frac{d}{dx}[\sin{(xy)}] + \sin{(xy)}\,\frac{d}{dx}(x + y) = 0


    To evaluate \frac{d}{dx}[\sin{(xy)}] you need to use the chain rule.

    Let u = xy so that the function becomes \sin{u}.

    \frac{du}{dx} = y + x\,\frac{dy}{dx}

    \frac{d}{du}(\sin{u}) = \cos{u} = \cos{(xy)}.


    Therefore \frac{d}{dx}[\sin{(xy)}] = \cos{(xy)}\left[y + x\,\frac{dy}{dx}\right].


    Go back to your original DE...

    (x + y)\,\frac{d}{dx}[\sin{(xy)}] + \sin{(xy)}\,\frac{d}{dx}(x + y) = 0

    (x + y)\cos{(xy)}\left[y + x\,\frac{dy}{dx}\right] + \sin{(xy)}\left[1 + \frac{dy}{dx}\right] = 0.


    Now try to solve for \frac{dy}{dx}.
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  3. #3
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    Hi

    Its probably obvious but I'm not sure how to go about solving for \frac{dy}{dx}<br />
    Thanks
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  4. #4
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    Quote Originally Posted by bobred View Post
    Hi

    Its probably obvious but I'm not sure how to go about solving for \frac{dy}{dx}<br />
    Thanks
    Distribute, move everything that has \frac{dy}{dx} to one side, factorise, solve for \frac{dy}{dx}.
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