Use the four-step process to find \( f^{\prime}(x) \) and then find \( f^{\prime}(-2), f^{\prime}(3) \), and \( f^{\prime}(7) \). \( f(x)=2 x^{2}+x-1 \) \( f^{\prime}(x)=\square \) \( f^{\prime}(-2)=\square \) (Type an integer or a simplified fraction.) \( f^{\prime}(3)=\square \) (Type an integer or a simplified fraction.) \( f^{\prime}(7)=\square \) (Type an integer or a simplified fraction.)

To find the derivative \( f^{\prime}(x) \) of the function \( f(x) = 2x^{2} + x - 1 \) using the four-step process, follow these steps: ### Step 1: State the Function \[ f(x) = 2x^{2} + x - 1 \] ### Step 2: Apply the Power Rule to Each Term The power rule states that if \( f(x) = ax^{n} \), then \( f^{\prime}(x) = a \cdot n \cdot x^{n-1} \). - For the term \( 2x^{2} \): \[ \frac{d}{dx}(2x^{2}) = 2 \cdot 2x^{2-1} = 4x \] - For the term \( x \) (which is \( x^{1} \)): \[ \frac{d}{dx}(x) = 1 \cdot 1x^{1-1} = 1 \] - For the constant term \( -1 \): \[ \frac{d}{dx}(-1) = 0 \] ### Step 3: Combine the Derivatives Add the derivatives of each term to get the derivative of the entire function: \[ f^{\prime}(x) = 4x + 1 \] ### Step 4: Simplify The derivative is already simplified: \[ f^{\prime}(x) = 4x + 1 \] ### Evaluating the Derivative at Specific Points 1. **At \( x = -2 \):** \[ f^{\prime}(-2) = 4(-2) + 1 = -8 + 1 = -7 \] 2. **At \( x = 3 \):** \[ f^{\prime}(3) = 4(3) + 1 = 12 + 1 = 13 \] 3. **At \( x = 7 \):** \[ f^{\prime}(7) = 4(7) + 1 = 28 + 1 = 29 \] ### Final Answers - \( f^{\prime}(x) = 4x + 1 \) - \( f^{\prime}(-2) = -7 \) - \( f^{\prime}(3) = 13 \) - \( f^{\prime}(7) = 29 \)