Calculus help..Find by implicit differentiation?

   5x² + 5x + xy = 3  ...  differentiate implicitly

 10x + 5 + xý + y = 0

        ý = - (10x + y + 5) ⁄ x

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 Find ý(x) = ý(3) : ... y_coordinate not given.

        ý(3) = - [(10)(3) + y + 5)] ⁄ 3

        ý(3) = - (y + 35) ⁄ 3

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 You can solve for "y" in the initial equation:

  5x² + 5x + xy = 3

       y = (3 − 5x² − 5x) ⁄ x

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    ý = - (y + 35) ⁄ 3  ...  and substitute for "y"

    ý = - ( [ (3 − 5x² − 5x) ⁄ x ] + 35 )   ⁄   3

 ý(3) = - ( [ (3 − 45 − 15) ⁄ 3 ] + 35 )   ⁄   3

 ý(3) = - 16 ⁄ 3

In implicit differentiation you assume y is a differentiable function of x. Instead of solving for y, take derivatives of the equation using the chain rule, and afterwards solve for y'(x) for the value of x you are interested in. This can be much easier than finding the derivative of a complicated expression for y.

Given:

5 x^2 + 5 x + x y(x) = 3 (Differentiate this)

y(3) = -19 (Save this for later)

10 x + 5 + (x y'(x) + 1 y(x)) = 0 (Took derivative of each term, considering y a function of x)

That was easy - now substitute x = 3 and y=-19 but y'(3) is still the one unknown quantity.

10 (3) + 5 + 3 y'(3) + 1 (-19) = 0

30 + 5 + 3 y'(3) - 19 = 0

3 y'(3) = 19 - 5 - 30 = -16

y'(3) = -16/3 if I did the arithmetic right.

10x + 5 + xy' + y = 0

y' = (-y - 5 - 10x)/x

To find the y value corresponding to x = 3:

45 + 15 + 3y = 3

y = -19

y'(3) = (19 - 5 - 30)/3 = -16/3