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------------------------------------------------ Find [tex]f'(x)[/tex] if [tex]f(x) = 3x^5(x^4 - 3)[/tex].

A. [tex]f'(x) = 45x^8 - 108x^5[/tex]
B. [tex]f'(x) = 45x^8 - 15x^5[/tex]
C. [tex]f'(x) = 45x^9 - 108x^5[/tex]
D. [tex]f'(x) = 45x^9 - 15x^5[/tex]

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.