# How do you differentiate #f(x)=xsqrt(abs(x-1)# using the product rule?

##### 1 Answer

Please see the explanation section below.

#### Explanation:

Because it involves the absolute value function, we need to first analyze the function.

Note that

So we have

Now we can differentiate using

# = (2x-2+x)/(2sqrt(x-1)) = (3x-2)/(2sqrt(x-1))#

and

# = (2-2x-x)/(2sqrt(1-x)) = (2-3x)/(2sqrt(1-x))# .

Putting these together, we have

**Additional comments**

We can write the derivative of

Or, we can use

With this notation, we get

# = 1/(2sqrt(abs(x-1))) abs(x-1)/(x-1)# .

We can write