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Wiskomen
Hi everyone, I'm still at high school and the level of maths there is half of the level of maths of the Single Variable Calculus course I'm following now, so there are sometimes symbols I don't understand... What does the symbol | mean in this question: The Derivative of |x| The slope of the graph of f(x)= |x| changes abruptly when x = 0. Does this function have a derivative? If so, what is it? If not, why not? Thank you! Wiskomen.
|x| means the absolute value of x. For example, |1| (absolute value of 1) is 1 and |-1| (absolute value of negative 1) is 1. That means that the slope of the graph f(x) = |x| is always 1 when x > 0, and always -1 when x < 0 but when x = 0 the slope is 0. You can see this when you look at the graph of f(x) = |x|. This video shows an example of how to get the derivative of an absolute value function. http://www.youtube.com/watch?v=sdoBDhcEe5M
While the above explanation of the absolute value is correct it should be noted that the absolute value function: \[f(x) = \left| x \right|\] is in fact not differentiable at x=0. And therefore in its entirety not differentiable. For a function to be differentiable at a certain point x0 the limits of the derivative of the function to the left side and the right side of this point have to be the same. So: \[\lim_{x \rightarrow (x_0)^-} f \prime(x) = \lim_{x \rightarrow (x_0)^+} f \prime (x)\] In this case the limit of the lefthand derivative as x->0 is -1 and the limit of the righthand derivative is 1. Since they are not the same the function is not differentiable at x=0.
Thank you soap and MattBenjamins for you answers! I understand now what | means. Thanks again! Wiskomen.