# 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 $(0,-3)$.