Calc2.44

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Find the slope of the tangent line to the graph x^{2}+y^{2}=9 at the point (0,-3).

Here, use implicit differentiation to find y'. Then plug in our point to find the slope of the tangent line at that specific point. By taking the derivative of both sides we get

2x+2yy'=0\,

Some students are tempted to plug in the point before taking the derivative. However, x, for example, is a variable and 0 is a constant. Thus, plugging in 0 before taking the derivative will give us a totally different derivative. After taking the derivative, it is okay to plug in the point. We have two options. We can solve for y' and then plug in our point or we can plug our point and then solve for y'. Usually, it is simpler to simply plug in the point. Here, this gives

0+2(-3)y'=0\,

or

y'=0\,

So, if we were to draw the tangent line to this graph, a circle, we would get a horizontal line through the point (0,-3).


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