Find the slope of the curve x^2 + xy + y^2 = 7 at (1,2)

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x^2 + xy + y^2 = 7

The slope is the derivative at (1,2).

Let us diffirentiate with respect to x:

=(x^2 + xy + y^2)' = (7)'

=(x^2)' + (xy)' + (y^2)' = (7)'

= 2x + (y' + y) + 2yy' = 0

==> 2x + y' +...

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x^2 + xy + y^2 = 7

The slope is the derivative at (1,2).

Let us diffirentiate with respect to x:

=(x^2 + xy + y^2)' = (7)'

=(x^2)' + (xy)' + (y^2)' = (7)'

= 2x + (y' + y) + 2yy' = 0

==> 2x + y' + y + 2yy' = 0

==> Now substitue with (1,2)

==> 2*1 + y' + 2 + 2*2*y' = 0

==> 2 + y' + 2 + 4y' = 0

==> 4 + 5y' = 0

==> 5y' = -4

==> y' = -4/5

Then the slope is -4/5

 

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