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Math Help - dy/dx of a polynomial in x and y

  1. #1
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    dy/dx of a polynomial in x and y

    what is dy/dx of x^2 + sin(xy) +y^2 = 0?

    I would guess the answer is

    2x + cos (xy)* y = 0

    Is this correct?

    MH
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    Re: dy/dx of a polynomial in x and y

    The derivation would be: 2x + cos (xy) [ 1 + x dy/dx ] + 2y dy/dx = 0
    that is: 2x + cos ( xy ) + x cos(xy) dy/dx + 2y dy/dx = 0
    now you can find dy/dx
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  3. #3
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    Re: dy/dx of a polynomial in x and y

    Thanks! Why is it dy/dx of sin(xy) = cos(xy)[1+ x dy/dx ]? I would have thought cos(xy)[y + x dy/dx]
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    Re: dy/dx of a polynomial in x and y

    Quote Originally Posted by kingsolomonsgrave View Post
    what is dy/dx of x^2 + sin(xy) +y^2 = 0?

    I would guess the answer is

    2x + cos (xy)* y = 0

    Is this correct?

    MH
    Are we assuming that y is a function of x?
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    Re: dy/dx of a polynomial in x and y

    You are right it was a typing error. On differentiating wrt x we get
    2x + sin (xy) [ y + x dy/dx] + 2y dy/dx]

    We foffow chain rule for sin(xy): d/dx(sin (xy) ) = cos ( xy) d/dx(xy) = cos xy [ y + x dy/dx ]
    THANKS
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    Re: dy/dx of a polynomial in x and y

    yes I think we are to assume y is a function of x
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    Re: dy/dx of a polynomial in x and y

    Quote Originally Posted by kingsolomonsgrave View Post
    Thanks! Why is it dy/dx of sin(xy) = cos(xy)[1+ x dy/dx ]? I would have thought cos(xy)[y + x dy/dx]
    First, please stop saying "dy/dx of" something. You are asking for dy/dx itself, where y is (implicitely) defined by that equation.

    However yes, the derivative of sin(u) with respect to x is cos(u)(du/dx). And the derivative of u= xy, with respect to x, is (dx/dx)y+ x(dy/dx)= y+ xdy/dx so that the derivative of sin(xy), with respect to x, is cos(xy)(y+ x dy/dx).
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    Re: dy/dx of a polynomial in x and y

    thanks HallsofIvey. Why wouldn't I say 'dy/dx of'? I thought dy/dx = the derivative?

    Or is it that dy/dx is read 'the derivative of y wrt x' and so contains 'the derivative' in its definition?
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    Re: dy/dx of a polynomial in x and y

    The derivative is an expression, ie a function. What you are referring to (in English) is taking the derivative of an expression. The two are vastly different.

    -Dan
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    Re: dy/dx of a polynomial in x and y

    You could say " \frac{d}{dx} of a function" (or, more generally, of an expression or equation). Note that there is no "y" in that. It is the function that takes the place of the 'y'.
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  11. #11
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    Re: dy/dx of a polynomial in x and y

    This is because \displaystyle \begin{align*} \frac{d}{dx} \end{align*} is the OPERATOR, while \displaystyle \begin{align*} \frac{dy}{dx} = \frac{d}{dx} \left( y \right) \end{align*}, the RESULT after the operator has operated on the function y.
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