Question

e) \\( y = 2x(3x^4 - x) \\)

\\( u = 2x \\)
\\( u = 2 \\)

\\( v = 3x^4 - x \\)
\\( v = 12x^3 \\)

\\( y = 2x(12x^3) + 2(3x^4 - x) \\)
\\( = 24x^4 + 6x^4 - 2x \\)
\\( = 30x^4 - 2x \\)

Explanation:

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<reasoning>

Identify the given function and components

Using the Product Rule and Polynomial Derivative knowledge points
\[

$$\begin{aligned} &y = 2x(3x^4 - x)\\ &u = 2x \implies u' = 2\\ &v = 3x^4 - x \implies v' = 12x^3 - 1 \end{aligned}$$

\]

Analyze the student's work for errors

Using the Power Rule and Polynomial Derivative knowledge points
\[

$$\begin{aligned} &\text{Student's } v' = 12x^3 \quad (\text{Error: forgot to differentiate the } -x \text{ term})\\ &\text{Student's } y' \text{ setup: } 2x(12x^3) + 2(3x^4 + x) \quad (\text{Error: wrote } +x \text{ instead of } -x \text{ in } v) \end{aligned}$$

\]

Calculate the correct derivative

Using the Product Rule and Power Rule knowledge points
\[

$$\begin{aligned} y' &= u'v + uv'\\ y' &= 2(3x^4 - x) + 2x(12x^3 - 1)\\ y' &= 6x^4 - 2x + 24x^4 - 2x\\ y' &= 30x^4 - 4x \end{aligned}$$

\]
</reasoning>

<answer>
The student made two errors in their work:

1. In finding \(v'\), they differentiated \(3x^4\) correctly to \(12x^3\) but forgot to differentiate the \(-x\) term (which should be \(-1\)), resulting in an incorrect \(v' = 12x^3\).
2. In the first line of the \(y'\) expansion, they incorrectly changed the sign of the last term in \(v\) from minus to plus, writing \(2(3x^4 + x)\) instead of \(2(3x^4 - x)\).

The correct derivative is:
\[y' = 30x^4 - 4x\]
</answer>

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Answer:

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<reasoning>

Identify the given function and components

Using the Product Rule and Polynomial Derivative knowledge points
\[

$$\begin{aligned} &y = 2x(3x^4 - x)\\ &u = 2x \implies u' = 2\\ &v = 3x^4 - x \implies v' = 12x^3 - 1 \end{aligned}$$

\]

Analyze the student's work for errors

Using the Power Rule and Polynomial Derivative knowledge points
\[

$$\begin{aligned} &\text{Student's } v' = 12x^3 \quad (\text{Error: forgot to differentiate the } -x \text{ term})\\ &\text{Student's } y' \text{ setup: } 2x(12x^3) + 2(3x^4 + x) \quad (\text{Error: wrote } +x \text{ instead of } -x \text{ in } v) \end{aligned}$$

\]

Calculate the correct derivative

Using the Product Rule and Power Rule knowledge points
\[

$$\begin{aligned} y' &= u'v + uv'\\ y' &= 2(3x^4 - x) + 2x(12x^3 - 1)\\ y' &= 6x^4 - 2x + 24x^4 - 2x\\ y' &= 30x^4 - 4x \end{aligned}$$

\]
</reasoning>

<answer>
The student made two errors in their work:

1. In finding \(v'\), they differentiated \(3x^4\) correctly to \(12x^3\) but forgot to differentiate the \(-x\) term (which should be \(-1\)), resulting in an incorrect \(v' = 12x^3\).
2. In the first line of the \(y'\) expansion, they incorrectly changed the sign of the last term in \(v\) from minus to plus, writing \(2(3x^4 + x)\) instead of \(2(3x^4 - x)\).

The correct derivative is:
\[y' = 30x^4 - 4x\]
</answer>

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