Question

find the derivative of the function ( g(x) = (x^2 - 5x - 4)e^x ) ( g(x) = )

Explanation:

Step1: Identify the product rule

The function \( g(x) = (x^2 - 5x - 4)e^x \) is a product of two functions, \( u(x)=x^2 - 5x - 4 \) and \( v(x)=e^x \). The product rule states that \( (uv)' = u'v + uv' \).

Step2: Find \( u'(x) \)

Differentiate \( u(x)=x^2 - 5x - 4 \) with respect to \( x \). Using the power rule, \( u'(x) = 2x - 5 \).

Step3: Find \( v'(x) \)

Differentiate \( v(x)=e^x \) with respect to \( x \). The derivative of \( e^x \) is \( e^x \), so \( v'(x) = e^x \).

Step4: Apply the product rule

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the product rule:
\[

$$\begin{align*} g'(x) &= (2x - 5)e^x + (x^2 - 5x - 4)e^x \\ &= e^x[(2x - 5) + (x^2 - 5x - 4)] \\ &= e^x(x^2 - 3x - 9) \end{align*}$$

\]

Answer:

\( g'(x) = (x^2 - 3x - 9)e^x \)