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:

Identify the given function and components

Using the Product Rule and Polynomial Derivative knowledge points

$$
LATEXBLOCK0
$$

Analyze the student's work for errors

Using the Power Rule and Polynomial Derivative knowledge points

$$
LATEXBLOCK1
$$

Calculate the correct derivative

Using the Product Rule and Power Rule knowledge points

$$
LATEXBLOCK2
$$

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:

$$y' = 30x^4 - 4x$$