Differentiate f(x) = 2x^2 * ln x

Given the function f(x) = 2x^2 * ln
x.

We need to determine the first
derivative.

We notice that the function f(x) is a product
of two functions.

Then, we will use the product rule to
find the derivative.

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

==> u= 2x^2  ==>   u' =
4x

==> v = ln(x)   ==>    v' =
1/x

Then we know that:

f'(x) =
u'*v + u*v'

Let us
substitute.

==> f'(x) = 4x*lnx + 2x^2 *
1/x

==> f'(x) = 4x*lnx +
2x

==> We will factor 2x from both
terms.

==> f'(x) = 2x*( 1 + 2lnx
).