Differentiate. g(x) = (x^2*cosx - cosx)

Given the function g(x) = (x^2*cosx -
cosx).

We need to differentiate the function
g(x).

First we will simplify the
function.

==> g(x) = x^2*cosx -
cosx

We notice that cosx is a common factor for both
terms.

Then we will factor
cosx.

==> g(x) = cosx( x^2 -
1)

Now we will use the product rule to find the
derivative.

Let g(x) = u*v such
that:

u= cosx  ==>   u' =
-sinx

v= x^2 -1  ==> v' =
2x

==> g'(x) = u'*v +
u*v'

                = -sinx(x^2-1) +
cosx*2x

                = 2x*cosx - x^2*sinx +
sinx

==> g'(x) = 2x*cosx - x^2*sinx +
sinx.