How do you use implicit differentiation to find the slope of the curve given #x^2+xy+y^2=7# at (3,2)?

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Answer 1 · 2016-10-26T02:53:19

See below.

This point is not actually on the graph. Let's pick the point #(3, -1)#.

We start by differentiating, using the product rule, the power rule and implicit differentiation.

#2x + y + x(dy/dx) + 2y(dy/dx) = 0#

#x(dy/dx) + 2y(dy/dx) = -y - 2x#

#dy/dx(x + 2y) = -y - 2x#

#dy/dx= (-y - 2x)/(x+ 2y)#

The slope of the curve is given by evaluating the point within the derivative.

So, letting the slope be #m#, we have:

#m = (-(-1) - 2(3))/(3 + 2(-1))#

#m = (1 - 6)/(3 - 2)#

#m = -5#

Hopefully this helps!