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# Derivative of Absolute value watch

1. How do you find a derivative when part of the function is in absolute value?
ex-
f(x)= 3(x-1)^(2/3) +|2x-1|
2. If x is not 1/2, you can rewrite this without the absolute value (i.e. if x > 1/2, |2x-1| = 2x-1, if x < 1/2, |2x-1| = 1-2x).

f is not going to be differentiable at x = 1/2. (think of the shape of |2x-1| near x=1/2).
3. (Original post by PatchworkTeapot)
How do you find a derivative when part of the function is in absolute value?
ex-
f(x)= 3(x-1)^(2/3) +|2x-1|
d/dx|x| = sgn(x), x cannot be zero
4. (Original post by DFranklin)
If x is not 1/2, you can rewrite this without the absolute value (i.e. if x > 1/2, |2x-1| = 2x-1, if x < 1/2, |2x-1| = 1-2x).

f is not going to be differentiable at x = 1/2. (think of the shape of |2x-1| near x=1/2).
That makes sense,but how would you come up with an overall derivative for this function, taking the 2 cases into account?
5. (Original post by PatchworkTeapot)
That makes sense,but how would you come up with an overall derivative for this function, taking the 2 cases into account?
I'd just write the two cases.

If you really want to do it in one go, you can write the derivative of |x| as x / |x| (note it's still indeterminate at x = 0). Similarly for derivative of |ax-b| etc...
6. (Original post by PatchworkTeapot)
That makes sense,but how would you come up with an overall derivative for this function, taking the 2 cases into account?
What's wrong with a piecewise-defined function?

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