Questions: In Exercises 19-40, calculate the derivative. 19. f(x)=(x^3+5)(x^3+x+1) 20. f(x)=(1/x-x^2)(x^3+1)

Solution Steps

To calculate the derivative of the given functions, we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by: \[ (u \cdot v)' = u' \cdot v + u \cdot v' \]

For each function, we will:

1. Identify \( u(x) \) and \( v(x) \).
2. Compute the derivatives \( u'(x) \) and \( v'(x) \).
3. Apply the product rule to find the derivative of the product.

To find the derivative \( f'(x) \), we apply the product rule: \[ f'(x) = u'v + uv' \] where \( u = x^3 + 5 \) and \( v = x^3 + x + 1 \).

Calculating the derivatives:

• \( u' = 3x^2 \)
• \( v' = 3x^2 + 1 \)

Substituting these into the product rule: \[ f'(x) = (3x^2)(x^3 + x + 1) + (x^3 + 5)(3x^2 + 1) \]

Again, we apply the product rule: \[ f'(x) = u'v + uv' \] where \( u = \frac{1}{x} - x^2 \) and \( v = x^3 + 1 \).

Calculating the derivatives:

• \( u' = -2x - \frac{1}{x^2} \)
• \( v' = 3x^2 \)

Substituting these into the product rule: \[ f'(x) = (3x^2)\left(\frac{1}{x} - x^2\right) + \left(-2x - \frac{1}{x^2}\right)(x^3 + 1) \]

Final Answer