Use the rules 𝑑 𝑑𝑥 cos(𝑥) = − sin(𝑥) and 𝑑 𝑑𝑥 cos(𝑓(𝑥)) = −𝑓′(𝑥) sin(𝑓(𝑥)).

Find the velocity of a particle at $t = 3$ seconds if its displacement (in metres) from a reference point is given by $s(t) = 12\cos\left(\frac{\pi t}{6}\right)$, where $t$ is time in seconds.

Determine the derivative of f(x) = 3cos(2x² - 5). Express your answer in simplest form.

Find the derivative of $y = \cos(2x)$ with respect to $x$.

A particle moves along a straight line. Its displacement from the origin at time \(t\) seconds is given by \(s(t) = 8\cos(3t^2 + 2)\) metres, where \(t \geq 0\). (a) Use the chain rule to determine an expression for the velocity \(v(t)\) of the particle at time \(t\). (b) Use the expression from part (a) to calculate the velocity of the particle when \(t = 1.5\) seconds. Give your answer in metres per second, correct to two decimal places.

Find $\frac{dy}{dx}$ for each of the following functions. (a) $y = \cos(3x)$ [1 mark] (b) $y = 5\cos(2x) - 4$ [1 mark] (c) $y = x^2\cos(x)$ [1 mark]