Use the rules 𝑑 𝑑𝑥 𝑒𝑥 = 𝑒𝑥 and 𝑑 𝑑𝑥 𝑒𝑓(𝑥) = 𝑓′(𝑥) 𝑒𝑓(𝑥).

A pharmaceutical company is developing a new drug. The concentration \(C(t)\) of the drug in the bloodstream, measured in milligrams per litre, is modelled by \[ C(t) = 15t e^{-0.4t} \] where \(t\) is the time in hours after administration, \(0 \le t \le 12\). (a) Use differentiation rules to determine \(C'(t)\), the rate of change of concentration with respect to time. (2 marks) (b) Hence find the time at which the concentration reaches its maximum value. (2 marks) (c) Determine the maximum concentration of the drug in the bloodstream, correct to two decimal places. (1 mark)

Find the derivative of $y = e^{3x - 2}$ with respect to $x$.

A population of microorganisms grows according to the function $P(t) = 2{,}500e^{0.15t}$, where $t$ is the time in hours since the culture was started. (a) Find $\frac{dP}{dt}$. [1 mark] (b) Calculate the rate of population growth at $t = 4$ hours. [1 mark] (c) Use your answer to part (a) to determine how long it takes for the rate of growth to reach 1{,}000 organisms per hour. [1 mark]

The velocity of a particle (in m/s) at time $t$ seconds is given by $v(t) = 3e^{2t} - 5$. Which expression represents the acceleration of the particle?

Consider the function \( h(x) = e^{3x^2 - 5x} \). Determine \( h'(x) \).