Question 6
Determine the derivative of $y=2x\cos(3x)$
That’s it. Step through the hints below to see why. In 2024, 81.1% of Queensland students chose (A).
Not quite. The derivative of $\cos(3x)$ is $-3\sin(3x)$, with a minus sign. Drop that sign and $2x\times\left(-3\sin(3x)\right)$ turns into $+6x\sin(3x)$. In 2024, 7.7% of Queensland students chose (B).
Not quite. $-6\sin(3x)$ is $2\times\left(-3\sin(3x)\right)$: the derivatives of the two factors multiplied together. A product needs the product rule, $u'v+uv'$. In 2024, 8.5% of Queensland students chose (C).
Not quite. $-2\sin(3x)$ is $2\times\left(-\sin(3x)\right)$. That multiplies the derivatives of the two factors and also loses the 3 from the chain rule. Use the product rule, with $\dfrac{d}{dx}\cos(3x)=-3\sin(3x)$. In 2024, 2.2% of Queensland students chose (D).
Option (A) matches the gradient of the tangent here. Move the point: it matches at every $x$.
At $x=0$ the term $6x\sin(3x)$ is 0, so (A) and (B) both give 2. One point can’t tell them apart. Move the point.
Option () happens to match at this point only. Move the point a little and it stops matching, so it isn’t the derivative.
Option () doesn’t match the gradient of the tangent here, so it can’t be the derivative.
$y=2x\cos(3x)$ is two functions of $x$ multiplied together. That calls for the product rule.
$$\frac{dy}{dx}=u'v+uv'$$
Let $u=2x$ and $v=\cos(3x)$.
The first factor is easy. The second needs the chain rule: the derivative of $\cos$ is $-\sin$, then multiply by the derivative of $3x$, which is 3.
$$u'=2$$
$$v'=-3\sin(3x)$$
Substitute into the product rule.
$\displaystyle \frac{dy}{dx}=2\cos(3x)$$\displaystyle +2x\left(-3\sin(3x)\right)$
$\displaystyle =2\cos(3x)-6x\sin(3x)$
Check with the figure at $x=\tfrac{\pi}{6}$. The tangent’s gradient is $-\pi\approx-3.142$. There (A) gives $-\pi$, (B) gives $\pi$, (C) gives $-6$ and (D) gives $-2$.
$$u=2x,\qquad v=\cos(3x)$$
$$u'=2,\qquad v'=-3\sin(3x)$$
$\displaystyle \frac{dy}{dx}=u'v+uv'$$\displaystyle =2\cos(3x)+2x\left(-3\sin(3x)\right)$
$\displaystyle =2\cos(3x)-6x\sin(3x)$
This is the product rule, with the chain rule inside it for the derivative of $\cos(3x)$.
How Queensland students answered in 2024
From the QCAA Mathematical Methods subject report for 2024. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2024 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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