Questions: Use the Product Rule to find the derivative of the function. f(x)=(8 x-x^3)(2 x+7) f'(x)=

Transcript text: Use the Product Rule to find the derivative of the function. \[ \begin{array}{l} f(x)=\left(8 x-x^{3}\right)(2 x+7) \\ f^{\prime}(x)= \end{array} \]

Solution

Solution Steps

To find the derivative of the function \( f(x) = (8x - x^3)(2x + 7) \) using the Product Rule, follow these steps:

Identify the two functions being multiplied: \( u(x) = 8x - x^3 \) and \( v(x) = 2x + 7 \).
Compute the derivatives of these functions: \( u'(x) \) and \( v'(x) \).
Apply the Product Rule: \( f'(x) = u'(x)v(x) + u(x)v'(x) \).

Step 1: Define the Functions

We start with the function \( f(x) = (8x - x^3)(2x + 7) \). We identify the two components:

\( u(x) = 8x - x^3 \)
\( v(x) = 2x + 7 \)

Step 2: Compute the Derivatives

Next, we compute the derivatives of \( u \) and \( v \):

\( u'(x) = 8 - 3x^2 \)
\( v'(x) = 2 \)

Step 3: Apply the Product Rule

Using the Product Rule, we find the derivative \( f'(x) \): \[ f'(x) = u'(x)v(x) + u(x)v'(x) \] Substituting the values, we have: \[ f'(x) = (8 - 3x^2)(2x + 7) + (8x - x^3)(2) \]

Step 4: Simplify the Expression

Now, we simplify the expression for \( f'(x) \): \[ f'(x) = (8 - 3x^2)(2x + 7) + 2(8x - x^3) \] Expanding this gives: \[ f'(x) = -2x^3 + 16x + (8 - 3x^2)(2x + 7) \] After further simplification, we arrive at: \[ f'(x) = -8x^3 - 21x^2 + 32x + 56 \]