What is the derivative of #|x|#?

Answer 1

In general, there is no derivative as #f(x) = |x|# is not a continuous function.

However, some may find it useful to find the derivative of the function by breaking it into pieces and kind of "avoiding" the discontinuity.

#f'(x) = 1 \text( ; when x > 0)#
#f'(x) = -1 \text( ; when x < 0)#
#f'(x) \text( is undefined ; when x = 0) #

In general, there is no derivative as #f(x) = |x|# is not a continuous function.

Finding the derivative of the function by dissecting it and essentially "avoiding" the discontinuity, however, might be helpful to some people.

The derivative is 1 because the positive side of the function is just a line with a positive slope of 1.

The derivative is -1 when looking at the function's negative side, which is just a line with a negative slope of 1.

Since the two one-side limits conflict at zero, there is obviously a discontinuity and no derivative can be defined.

So you could summarize a piecewise derivative of #f(x) = |x|# as: #f'(x) = 1 \text( ; when x > 0)# #f'(x) = -1 \text( ; when x < 0)# #f'(x) \text( is undefined ; when x = 0) #
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Answer 2

The derivative of |x| is not defined at x = 0. For x > 0, the derivative is 1, and for x < 0, the derivative is -1.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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