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
