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Show that the function $ f(x) = | x - 6 | $ is not differentiable at 6. Find a formula for $ f' $ and sketch its graph.

limits are not equal, $f^{\prime}(6)=\lim _{x \rightarrow 6} \frac{f(x)-f(6)}{x-6}$ does not exist and $f$ is not differentiable at 6

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So we are looking at the function f of x equals the absolute value of X -6. Uh And here is a graph of that function. So just blue graph Uh is the image of the function absolute value of X -6. For any uh X greater than six. Any X value greater than six. You are on this portion of the graph which is linear. It is a line and the slope of this line uh is one. So for any X value greater than six, uh the slope of the line is one. And for a line the slope of the line is going to be the derivative of the function if it's a linear function. So F prime of X. The derivative of this function equals one. Uh when X is greater than six. Okay, so for this portion of the X axis, the slope of the function is one. Since this linear. So the derivative of the function is one For X is greater than six. If we look at X values that are less than six, that's this portion of the X axis, Everything to the left of six. then this is the portion of the graph corresponding to X values that are less than six. You can clearly see that this portion of the graph is also linear and the slope of this portion of the graph is negative one. So f prime of X equals negative one. When X is less than six, this is supposed to be a comma f prime of X equals negative one. When X is less than six. Well, f is not differentiable at six because if f prime of x existed and was defined when x equals six uh than the derivative f prime of X to the right side of six. Uh And to the left side of six, uh would would have they would have to be approaching the derivative that prime of X as X approaches six from the positive side at prime of X Would have to exist. And it does it equals one. But as X approaches six from the negative side, uh F prime of X would also have to exist and it does equals negative one. But for F prime of X to exist at six, uh prime of X as X approaches from the positive side would have to be approaching the same limit as F prime of X. Uh Coming in from the negative side and they don't okay, on the positive side of six, at prime of X is one on the left side of six, F prime of X is negative one. So F prime of X is not going to be defined at six because when we approach from uh the positive And from the negative sides of six, the f prime of X approaches to different limits. The limit of F prime of X as X comes in from the positive side is one, the limit of F prime of X as X approaches from the negative side or the left side of six is negative one. So f prime of X at six is not defined. So next we actually want to define F prime of X. Uh and or find a formula for F prime of X and we actually already have it. Uh This fray here is the formula for F prime of X. The derivative of this function equals one when X is greater than six and equals negative one. When X is less than six. Last but not least. Uh We need to sketch the graph of F prime of X. Okay, so now we're going to graph this uh F prime of X function that prime of X equals one when X is greater than six. So when X is greater than six, f prime of X would equal one. So we're gonna put an open circle when X is six but for X greater than six, F prime of X equals one. When X is less than six to the left of six, that prime of X is going to equal negative one. So when X is less than six at prime of X equals negative one. Let's go ahead and get a line to do that for us. And once again ah we want we want an open circle here. Okay, so here is the graph of f prime of X. When X is greater in six. Okay, everything to the right of six, not including the six, that's why we have the open circle, F prime of X is one. Okay, so one on the vertical axis, S C F prime of X axis when X is less than six. So when we go to the left of six, that prime of X equals negative one. So for all X values to the left of six have prime of X is negative one. So here we are there once again at six. Um it's not included. Now notice once again we said that F prime of X was not able to be defined at six. That's why at six you have an open circle here. You have an open circle here and you do not have any other points for when X is exactly six because F prime of X is not defined when X equals six. Okay, as X approaches six from the right side, that prime of X uh is approaching one as X approaches six from the negative side from left side, that prime of X approaches the limit of negative one. Uh So you can't approach to different limit numbers. Uh when we come in from the two different sides uh and have a limit. So at prom index did not exist at six, that's why you do not have a point on the graph. Now, when X is six