Question

attempt 1: 10 attempts remaining. find the derivative ( y ) of the function ( y = (x^3 - 5x)e^{2x + 4} ). ( y = ) submit answer next item

Explanation:

Step1: Identify the product rule

The function \( y = (x^3 - 5x)e^{2x + 4} \) is a product of two functions, \( u = x^3 - 5x \) and \( v = e^{2x + 4} \). The product rule states that \( y' = u'v + uv' \).

Step2: Find \( u' \)

Differentiate \( u = x^3 - 5x \) with respect to \( x \). Using the power rule, \( u' = 3x^2 - 5 \).

Step3: Find \( v' \)

Differentiate \( v = e^{2x + 4} \) with respect to \( x \). Using the chain rule, the derivative of \( e^{f(x)} \) is \( e^{f(x)}f'(x) \). Here, \( f(x) = 2x + 4 \), so \( f'(x) = 2 \). Thus, \( v' = 2e^{2x + 4} \).

Step4: Apply the product rule

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

$$\begin{align*} y' &= (3x^2 - 5)e^{2x + 4} + (x^3 - 5x)(2e^{2x + 4}) \\ &= e^{2x + 4}(3x^2 - 5 + 2x^3 - 10x) \\ &= e^{2x + 4}(2x^3 + 3x^2 - 10x - 5) \end{align*}$$

\]

Answer:

\( y' = e^{2x + 4}(2x^3 + 3x^2 - 10x - 5) \)