Question

calculate the derivative of |1-x^2|

Answer 1

Have to do it piecewise, but pretty straight forward.

Answer 2

do you think you could show me step by step for each of the two?

Answer 3

Sure. First we break it up piecewise:
\[|a| = b \]\[\implies a = b \text{ if }a \ge 0 \text{ AND } a = -b \text{ if } a < 0\]
Therefore in this case we need to know, when is \(1-x^2\) positive, and when is it negative?

Answer 4

Then we can find the derivative will be the derivative of \(1-x^2\) on the interval where it's positive, and it will be the derivative of \(-(1-x^2) = x^2-1\) on the interval where it's negative. Where it is 0 it will not be differentiable.