What is the derivative of x|x|? and how

Since there's an absolute value in here, we need to define the equation in two parts (a piecewise function).

Let's start by saying f(x) = x|x|

Absolute value functions work different when the number inside the absolute value is positive from when it is negative.

If x is greater than or equal to 0, then |x| is the same as x. So, we have x|x| = x*x = x^2 for x >=0

When x is less than 0, then |x| is the same as -x.
For example, |-2| is the same as -(-2) = 2.
So, in this case, we have x|x| = x*-x = -x^2 for x < 0

So, our function is:

f(x) = x^2 if x >= 0 and -x^2 if x < 0

Now, we can find the derivative, f ', easily. Just take the derivative of each part separately.

So...

f '(x) = 2x if x >= 0 and -2x if x < 0


-Mike

It will be discontinuous at x=0

for x positive
this is the same as x^2, therefore derivative = 2x (which is always positive)

for x negative, this is the same as -x^2, therefore derivative = -2x (which is again positive)

therefore, the derivative is |2x|