Question

Find the derivative of f(x) = x|x|
How to solve this?

Answer 1

If x is negative, then what happens? If x is positive then what happens? Take the derivative of each case.

Answer 2

Can you do that?

Answer 3

Basically, \[
(x|x|)' = \left(\begin{cases} x^2 & x>0\\-x^2 & x<0\end{cases}\right)'=\begin{cases} 2x & x>0\\-2x & x<0\end{cases} = 2|x|
\]

Answer 4

Fortunately the function is differentiable at \(0\) which sometimes isn't the case when you have absolute values in a function.

Answer 5

very detailed, thank you so much ^^