Question

9 multiple choice 1 point find f(x) for the following function. f(x) = 2x(4x + 4)^5 2(4x + 4)^4 2(4x + 4)^5(9x + 4) 2(4x + 4)^4(24x + 4) 2(24x + 4)^4

Explanation:

Step1: Identify the rule to use

We need to find the derivative of \( f(x) = 2x(4x + 4)^5 \). This is a product of two functions, \( u = 2x \) and \( v=(4x + 4)^5 \), so we use the product rule \( (uv)'=u'v + uv' \). Also, for \( v=(4x + 4)^5 \), we need the chain rule.

Step2: Find \( u' \) and \( v' \)

• For \( u = 2x \), the derivative \( u'=\frac{d}{dx}(2x)=2 \).
• For \( v=(4x + 4)^5 \), using the chain rule: let \( t = 4x+4 \), so \( v = t^5 \). Then \( \frac{dv}{dt}=5t^4 \) and \( \frac{dt}{dx}=4 \). So \( v'=\frac{dv}{dx}=\frac{dv}{dt}\cdot\frac{dt}{dx}=5(4x + 4)^4\cdot4 = 20(4x + 4)^4 \).

Step3: Apply the product rule

Using \( (uv)'=u'v+uv' \):

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Answer:

\( 2(4x + 4)^4(24x + 4) \) (the third option)