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

1 Answer
Dec 23, 2015

Answer:

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))#