# Find the derivative of the function: f(x) = (2x − 3)^{4 }(x^{2} + 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 }(x^{2} + x + 1)^{5} is (2x - 3)^{3 }(x^{2} + x + 1)^{4} (28x^{2} - 12x - 7).

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

**Explanation: **

The derivative of x^{n} is nx^{n - 1}.

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

Derivative of (x^{2} + x + 1)^{5} is 5 (x^{2} + x + 1)^{4} × (2x + 1) = 5 (2x + 1) (x^{2} + x + 1)^{4}

Derivative of f(x) = (x^{2} + x + 1)^{5 }× Derivative of (2x − 3)^{4} + (2x − 3)^{4} × Derivative of (x^{2} + x + 1)^{5 }

= (x^{2} + x + 1)^{5} × 8 (2x − 3)^{3 }+ (2x − 3)^{4} × 5 (2x + 1) (x^{2} + x + 1)^{4}

= (2x - 3)^{3 }(x^{2} + x + 1)^{4} [8 (x^{2} + x + 1) + 5 (2x + 1) (2x − 3)]

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

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

= (2x - 3)^{3 }(x^{2} + x + 1)^{4} (28x^{2} - 12x - 7)

You can also use Cuemath's Online Derivative Calculator.

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

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