Find the derivative of the function: f(x) = (2x − 3)⁴ (x² + x + 1)⁵.

We will use the product rule to find the derivative of the given function.

Answer: The derivative of the function: f(x) = (2x − 3)⁴ (x² + x + 1)⁵ is (2x - 3)³ (x² + x + 1)⁴ (28x² - 12x - 7).

Let's use the product rule to find the answer.

Explanation: 

The derivative of xⁿ is nx^{n - 1}.

Derivative of (2x − 3)⁴ is 4 (2x − 3)³ × 2 = 8 (2x − 3)³

Derivative of (x² + x + 1)⁵ is 5 (x² + x + 1)⁴ × (2x + 1) = 5 (2x + 1) (x² + x + 1)⁴

Derivative of f(x) = (x² + x + 1)⁵ × Derivative of (2x − 3)⁴ + (2x − 3)⁴ × Derivative of (x² + x + 1)⁵ 

= (x² + x + 1)⁵ × 8 (2x − 3)³ + (2x − 3)⁴ × 5 (2x + 1) (x² + x + 1)⁴

= (2x - 3)³ (x² + x + 1)⁴ [8 (x² + x + 1) + 5 (2x + 1) (2x − 3)]

= (2x - 3)³ (x² + x + 1)⁴ [8x² + 8x + 8 + (10x + 5) (2x − 3)]

= (2x - 3)³ (x² + x + 1)⁴ [8x² + 8x + 8 + 20x² - 30x + 10x - 15]

= (2x - 3)³ (x² + x + 1)⁴ (28x² - 12x - 7)

You can also use Cuemath's Online Derivative Calculator.

Thus, the derivative of the function f(x) = (2x − 3)⁴ (x² + x + 1)⁵ is (2x − 3)³ (x² + x + 1)⁴ (28x² - 12x - 7).