Questions: Find the derivative of (f(x)). (f(x)=x^3 e^x) (f^prime(x)=square)

Transcript text: Find the derivative of $f(x)$. \[ \begin{array}{l} f(x)=x^{3} e^{x} \\ f^{\prime}(x)=\square \end{array} \]

Solution

Solution Steps

To find the derivative of the function $ f(x) = x^3 e^x $, we will use the product rule. The product rule states that if you have a function that is the product of two functions, say $ u(x) $ and $ v(x) $, then the derivative $ (uv)' $ is given by $ u'v + uv' $. Here, let $ u(x) = x^3 $ and $ v(x) = e^x $. We will find the derivatives $ u'(x) $ and $ v'(x) $, and then apply the product rule.

Step 1: Identify the Function

We are given the function $ f(x) = x^3 e^x $. To find its derivative, we will apply the product rule.

Step 2: Apply the Product Rule

Using the product rule, which states that $ (uv)' = u'v + uv' $, we identify: