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

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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

Sam E. A Level Maths tutor, GCSE Maths tutor

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