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Thread: Implicit Differentiation!!

  1. #1
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    Implicit Differentiation!!

    Find dy/dx.

    x^2 + xy - y^3 = xy^2

    can someone help me out with this one, im not sure exactly where to put dy/dx when differentiating. thanks

    mathlete
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  2. #2
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    Remember as you do these that y is a function of x, so when you have to differentiate y at any point, the result will be $\displaystyle \frac{dy}{dx}$ - simply the derivative of y with respect to x.

    $\displaystyle x^2 + xy - y^3 = xy^2$

    Take the derivative with respect to x on both sides.

    The derivative of $\displaystyle x^2 = 2x$.

    To find the derivative of xy, you need to use the product rule, and you get:

    $\displaystyle y + x\frac{dy}{dx}$.

    The derivative of $\displaystyle y^3 = 3y^2 \frac{dy}{dx}$.

    The derivative of $\displaystyle xy^2$ again requires the product rule, giving:

    $\displaystyle y^2 + 2xy \frac{dy}{dx}$.

    Put it all together and you get:

    $\displaystyle 2x + y + x\frac{dy}{dx} + 3y^2 \frac{dy}{dx} = y^2 + 2xy \frac{dy}{dx}$.

    Get all the dy/dx on one side, everything else on the other. Factor out dy/dx from all the terms that have it, then divide to get dy/dx by itself.
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