differentiating implicit function?

2012-05-21 10:49 am
Find the slope of tangent to the curve x^2 + y^2 = -xy+4y+11 at (3,-1)
solution:
d/dx(x^2 +y^2) = d/dx(-xy+4y+11)
2x + 2y(dy/dx) = -x(dy/dx) -y +4(dy/dx)
dy/dx = (-2x-y)/(x+2y-4)
Next step:substitude x=3,y=-1 into dy/dx.After that,the slope of tangent at (3,-1) will be found

I want to ask why dy/dx is the slope of tangent of the equation.
x^2 + y^2 = -xy+4y+11 is an implicit function,which cannot be expressed in terms of x,so why differentiating y with respect to x will give the slope of tangent?

回答 (2)

2012-05-21 11:06 am
✔ 最佳答案
The definition of dy/dx is "the rate of change of y with respect to x at the point (x, y)". Wherever that symbol appears in the equation, that's what it means, it doesn't change because you're differentiating implicitly. An implicit equation can be plotted, and at the point (x,y), its slope will be dy/dx.
2012-05-21 5:57 pm
Because it tells us the position of y with respect to x. This is the dictionary definition of "slope."

When differentiating implicitly, you could also throw in (dx/dx), but that's redundant, because it equals 1. You also could've differentiated x with respect to y, if you felt like having fun, but then your numbers would've been different. In a way, it's arbitrary. We choose to differentiate y with respect to x because that's what is most useful to us.


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