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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

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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 [tex]x\geq 0[/tex] 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

[tex]f'(x) = (x)' = 1 \text{if } x\geq 0[/tex]

[tex]f'(x) = (-x)' = -1 \text{if } x<0[/tex]

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

[tex]\lim_{x\to 0^+} f'(x) = 1[/tex] and

[tex]\lim_{x\to 0^-} f'(x) = -1[/tex]

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.

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