To find the derivative of the function \(f(x) = (x^2 + 8)(4x - 3)\) by first expanding the polynomials, you can use the product rule. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
First, expand the polynomials:
\(f(x) = (x^2 + 8)(4x - 3) = 4x^3 - 3x^2 + 32x - 24\)
Now, we can find the derivative of this expanded expression using the product rule:
\(f'(x) = (4x^3 - 3x^2 + 32x - 24)' = (4x^3)' - (3x^2)' + (32x)' - (24)'\)
Now, find the derivatives of each term separately:
1. \((4x^3)' = 12x^2\)
2. \((-3x^2)' = -6x\)
3. \((32x)' = 32\)
4. \((-24)'\ = 0\) (The derivative of a constant is always zero.)
Now, combine these derivatives using the product rule:
\(f'(x) = 12x^2 - 6x + 32 - 0\)
So, the derivative of the function \(f(x) = (x^2 + 8)(4x - 3)\) after expanding the polynomials is:
\(f'(x) = 12x^2 - 6x + 32\)