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lenorefrancis lenorefrancis
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12 years ago
Use implicit differentiation to find the slope of the tangent line to the curve

(3x^2)- (3xy) - (3y^3) = -3

at the point (1,1)
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#1
12 years ago
Implicit differentiation means that if you could solve an equation for y, it would equal some function of x.  So, every time you see a y, pretend that it is "hiding" a function of x.  Thus, you will need to use the chain rule on it.  

So, the derivative is:
6x -(3y+3x(dy/dx)) - 9y^2(dy/dx) = 0
notice, I had to use the product rule on 3xy

Solve for dy/dx:
6x - 3y - 3x(dy/dx) - 9y^2(dy/dx) = 0
-3x(dy/dx) - 9y^2(dy/dx) = 3y - 6x
factor out dy/dx:
(dy/dx)(-3x - 9y^2) = 3y - 6x
so,
dy/dx = (3y - 6x)/(-3x - 9y^2)

Now plug in your point where x=1 and y=1
dy/dx at (1,1) = (3 - 6)/(-3 - 9) = -3/-12 = 4


Hope this helped.
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