Use the product rule, 𝑑(𝑢𝑣) 𝑑𝑥 = 𝑢 𝑑𝑣 𝑑𝑥 + 𝑣 𝑑𝑢 𝑑𝑥, to determine the derivative of products of functions involving power and polynomial functions.

Find $\frac{d}{dx}\left[x^3(2x^2 - 5)\right]$ using the product rule.

A rectangular storage container is being designed such that its length \(L\) (in metres) varies with time \(t\) (in seconds) according to \(L(t) = 3t^2 + 2\), and its width \(W\) (in metres) varies according to \(W(t) = 4t^3 - 5t\). (a) Use the product rule to determine an expression for \(\frac{dA}{dt}\), where \(A(t)\) is the area of the base of the container. (b) Hence, determine the rate of change of the base area at \(t = 2\) seconds. Express your answer in square metres per second.

A particle moves along a straight line. Its displacement from a fixed origin at time \( t \) seconds is given by \( s(t) = (2t^2 - 5t)(3t^3 + 4t) \) metres, where \( t \geq 0 \). (a) Use the product rule to determine an expression for the velocity \( v(t) \) of the particle at time \( t \). (b) Hence determine the velocity of the particle when \( t = 2 \) seconds. Express your answer in metres per second, correct to one decimal place. (c) Use the information from part (a) to determine the time(s) when the particle is instantaneously at rest during the first 3 seconds of motion.