Derivative with an absolute value?

Question

find dy/dx of

y = |x^2 - 4|

Answer 1

The absolute value rule (or one formation of it) says that, if u is a function of x, then

du / dx = (u / |u|) * u'.

So, in this case, we have

dy/dx = ((x^2 - 4) / |x^2 - 4|) * 2x.

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EDIT: You're not allowed to use the chain rule? Then I guess you have to split up |x^2 - 4| into three pieces based on where it's positive and where it's negative:

y = x^2 - 4, if x < -2

y = 4 - x^2, if -2 <= x < 2

y = x^2 - 4, if x >= 2

y' = 2x, if x < -2

y' = -2x, if -2 < x < 2

y' = 2x, if x > 2.

The derivative does not exist at x = -2 or x = 2, because the graph suddenly switches direction there.

Answer 2

x>2

y=x^2 - 4

y' = 2x

x<2

y = 4-x^2

y' = -2x

undefined at x = 2

Or you can use the absolute value rule from above which give you the same thing

dy/dx = ((x^2 - 4) / |x^2 - 4|) * 2x.