Video Example EXAMPLE 5 Where is the function f(x) = |x| differentiable? SOLUTION If x > 0, then |x| = and we can choose h small enough that x + h > 0 and hence |x + h| = . Therefore, for x > 0 we have

Step-by-step explanation:

We say that a function is differentiable if it's derivative exists and it is continous over the domain of the original function. Recall that the function |x| is given as |x| = x if and |x|=-x if x<0. In both cases, we have either the function x or -x, which are one-degree polynomials. So, we can find easily their derivatives. So if f(x) = |x|. Then, we have that

We should check what happens x=0, since that is the point where the definition of f changes. We can check that

and

since this limits are not equal, the derivative is not continous at x=0. Hence, the derivative doesn't exist at x=0.

In this case, we say that |x| is differentiable at any x different from 0.

this shows a reflection over the x-axis and is translated one unit down and 6 units to the right (x + 6, y - 1)

hope this !

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step-by-step explanation:

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