Questions: Find (f^prime(x)) if [ f(x)=(x^2+6 x) e^x ] (Enter an exact answer. Use symbolic notation and fractions where needed.)

$Find (f^prime(x)) if [ f(x)=(x^2+6 x) e^x ] (Enter an exact answer. Use symbolic notation and fractions where needed.)$

Transcript text: Find $f^{\prime}(x)$ if \[ f(x)=\left(x^{2}+6 x\right) e^{x} \] (Enter an exact answer. Use symbolic notation and fractions where needed.)

Solution

Solution Steps

To find the derivative $ f^{\prime}(x) $ of the function $ f(x) = (x^2 + 6x) e^x $, we will use the product rule. The product rule states that if you have a function $ f(x) = u(x) \cdot v(x) $, then the derivative $ f^{\prime}(x) = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) $. Here, let $ u(x) = x^2 + 6x $ and $ v(x) = e^x $. We will find the derivatives $ u^{\prime}(x) $ and $ v^{\prime}(x) $, and then apply the product rule.

Step 1: Define the Function

We start with the function defined as: \[ f(x) = (x^2 + 6x) e^x \]

Step 2: Apply the Product Rule

To find the derivative $ f^{\prime}(x) $, we apply the product rule: \[ f^{\prime}(x) = u^{\prime}(x) v(x) + u(x) v^{\prime}(x) \] where $ u(x) = x^2 + 6x $ and $ v(x) = e^x $.

Step 3: Calculate the Derivatives

We compute the derivatives: \[ u^{\prime}(x) = 2x + 6 \] \[ v^{\prime}(x) = e^x \]

Step 4: Substitute into the Product Rule

Substituting the derivatives back into the product rule gives: \[ f^{\prime}(x) = (2x + 6)e^x + (x^2 + 6x)e^x \]

Step 5: Factor the Expression

We can factor out $ e^x $: \[ f^{\prime}(x) = e^x \left( (2x + 6) + (x^2 + 6x) \right) \] This simplifies to: \[ f^{\prime}(x) = e^x \left( x^2 + 8x + 6 \right) \]