How do you differentiate #f(x)=xsqrt(abs(x-1)# using the product rule?
Please see the explanation section below.
Because it involves the absolute value function, we need to first analyze the function.
So we have
Now we can differentiate using
# = (2x-2+x)/(2sqrt(x-1)) = (3x-2)/(2sqrt(x-1))#
# = (2-2x-x)/(2sqrt(1-x)) = (2-3x)/(2sqrt(1-x))#.
Putting these together, we have
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