Find the derivative of f(x) = (3x^2 - 8x)^3

Answer 1

If f(x) = (3x² - 8x)³, then taking the derivative would involve the power rule and the chain rule.

Step 1: To apply the power rule, we will do three things:

1. Multiply the exponent on the binomial (the 3) by the coefficient of the binomial (which is just a 1), and write their product in front of the parentheses as the new coefficient.
2. Leave everything inside of the parentheses alone (for now), completely untouched.
3. Reduce the original exponent by subtracting 1 from it (so 3-1).

This give us: 3 (3x² - 8x)² **We're not done yet though!

Step 2: Now we will need to apply the chain rule, by taking the derivative of the binomial inside of the parentheses:

Derivative of the binomial in parentheses = 6x - 8

Step 3: Multiplying the derivative from Step 2 by the expression that we got from Step 1, we get our final answer:

f'(x) = 3 (3x² - 8x)² (6x - 8)

Answer 2

It would be painful to expand the binomial by raising it to the 3rd power and then differentiating term by term, but it could be done. Easier to use the power rule followed by the chain rule.

f(x) = (3x² - 8x)³

f''(x) = 3 ( 3x² - 8x)² (6x - 8)