Answer :
Final answer:
To find the derivative of the function f(x) = 3x⁵(x⁴ - 3), the product rule is applied, followed by simplification which leads to the correct answer as option (c) f'(x) = 45x⁹ - 108x⁵.
Explanation:
The student has asked to find the derivative of the function f(x) = 3x⁵(x⁴ - 3). To do that, we first need to apply the product rule of differentiation since f(x) is a product of two functions, 3x⁵ and (x⁴ - 3).
The derivative of 3x⁵ is 15x⁴, and the derivative of (x⁴ - 3) is 4x³. Applying the product rule:
f'(x) = (3x⁵) * (4x³) + (x⁴ - 3) * (15x⁴)
Then we simplify:
f'(x) = 12x⁹ + 15x⁸ - 45x⁴
Factoring out 3x⁵, we get:
f'(x) = 3x⁵(4x⁴ + 5x³ - 15)
Now we inspect the given options:
- f'(x) = 45x⁸ - 108x⁵
- f'(x) = 45x⁸ - 15x⁵
- f'(x) = 45x⁹ - 108x⁵
- f'(x) = 45x⁹ - 15x⁵
Option (c) f'(x) = 45x⁹ - 108x⁵ is the correct answer after expanding the product and simplifying.
