find derivative `(sqrt(x))(x-1)` ? Would this just be `1/2x^(-1/2)` or would I need to apply the product rule? Could this be considered the Constant Multiple rule?
You must to use product rule. If you wish you can first multiply the expression like this
`sqrt x (x-1)=x sqrt x- sqrt x`
and then find the derivative, but this is not necessary, it only makes differentiation more difficult.
Let's now differentiate by using product rule.
`(sqrt x(x-1))'=sqrt x'(x-1)+sqrt x(x-1)'=(x-1)/(2sqrt x)+sqrt x`
you have two ways
`A)` first make products so to get:
`sqrt(x)(x-1)= xsqrt(x)-sqrt(x)` and then makes derivatres
`d/dx(xsqrt(x)-sqrt(x))=d/dx xsqrt(x) -d/dx sqrt(x)=` `d/dx x^(3/2) -d/dx x^(1/2)=`
`=3/2x^(3/2-1) -1/2 x^(1/2-1)= 3/2 x^(1/2)-1/2 x^(-1/2)` `=1/2(3sqrt(x)-1/sqrt(x))=`
`B)` Soon makes derivative of products:
`d/dx sqrt(x)(x-1)= (x-1) d/dx sqrt(x) + sqrt(x) d/dx (x-1) =`
`=1/2 (x-1)/sqrt(x) + sqrt(x)`
the two results look like differents, instead, still working on the last one :
`1/2 (x-1)/sqrt(x) +sqrt(x)= 1/(2sqrt(x))(x-1+2x)=sqrt(x)/(2x)(3x-1)`
The result is the same.