Final answer:
The derivative of the function f(x) = 3x²(6x² - 4x - 3) is found using the product rule and is f'(x) = 72x³ - 36x² - 18x.
Step-by-step explanation:
To find the derivative of the function f(x) with respect to x, where f(x) = 3x²(6x² - 4x - 3), we need to apply the product rule along with the power rule of differentiation.
The product rule states that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. In this case, our first function is 3x² and the second function is (6x² - 4x - 3).
Using the product rule:
- Differentiate the first function: The derivative of 3x² is 6x.
- Differentiate the second function: The derivative of (6x² - 4x - 3) is (12x - 4).
- Now apply the product rule: f'(x) = 3x² × (12x - 4) + (6x² - 4x - 3) × 6x.
Simplify to get the final derivative:
f'(x) = 36x³ - 12x² + 36x³ - 24x² - 18x
Combining like terms, we get:
f'(x) = 72x³ - 36x² - 18x