How would you differentiate f(x) = 2x(3x - 1)^2 using the chain rule?

In order to differentiate this expression, you need to use the chain rule. 

The chain rule gives: f'(x) = uv' + u'v.

The u and the v are two parts of the original function f(x). The apostrophe ' at the end means the derivative of that.

We need to assign values to u and v, so we look at the function f(x) = 2x(3x - 1)² to see what parts it is in:

u = 2x

v = (3x - 1)²

Then, we differentiate each of these.

u' = 2

v' = 2 x 3 x (3x - 1)¹ = 6(3x - 1)

Now, we can put this expression altogether:

f'(x) = uv' + u'v = 2x(6(3x - 1)) + 2(3x - 1)²

And now, simplify.

f'(x) = 12x(3x - 1) + 2(3x - 1)²

f'(x) = 2(3x - 1)[6x + (3x - 1)]

f'(x) = 2(3x - 1)(9x - 1)

f'(x) = 2(27x² - 9x - 3x + 1)

f'(x) = 2(27x² - 12x + 1)

f'(x) = 54x² - 24x + 2