Question

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

Answer 1

The point here is seeing that a root can be converted into a power, as stated by the following power rule: `a^(n/m)=root(m)(a^n)`

Explanation:

Using such concept, we can rewrite the expression: `f(x)=x(x-1)^(1/2)`

The product rule states that for a function `f(x)=g(x)h(x)`, then `f'(x)=g'(x)h(x)+g(x)h'(x)`

Before, let's also remember that the chain rule states `(dy)/(dx)=(dy)/(du)(du)/(dx)` and rename `u=x-1` so we can derivate the term `(x-1)^(1/2)`

`f'(x)=(1)(x-1)^(1/2)+x((1/(2u))(1))=`

`f'(x)=(x-1)^(1/2)+x/(2(x-1)^(1/2))=((x-1)+x)/(2(x-1)^(1/2))`

`f'(x)=(2x-1)/(2(x-1)^(1/2))`