Use the rules 𝑑 𝑑𝑥 ln(𝑥) = 1 𝑥 and 𝑑 𝑑𝑥 ln(𝑓(𝑥)) = 𝑓′(𝑥) 𝑓(𝑥).

The height, \( h \) (metres), of water in a reservoir at time \( t \) (days) is modelled by \( h(t) = 12 + 3\ln(2t + 5) \), where \( t \geq 0 \). State the rule used to differentiate \( h(t) \) with respect to \( t \), and hence determine an expression for \( h'(t) \).

The temperature \(T\) (in degrees Celsius) of a chemical reaction \(t\) minutes after it begins is modelled by \[T(t) = 85 + 45\ln(2t + 1)\] where \(0 \leq t \leq 30\). (a) Use an appropriate rule to determine the rate of change of temperature at \(t = 4\) minutes. (2 marks) (b) At a particular time during the reaction, the rate of change of temperature is exactly \(3\) degrees Celsius per minute. Use the information from part (a) to determine this time. (3 marks)

Differentiate \( f(x) = \ln(5x^2 + 3) \) with respect to \( x \).

The function \(h(x) = \ln(3x^2 + 5)\) is defined for \(x \in \mathbb{R}\). Which of the following expressions represents \(h'(x)\)?

Find $\frac{dy}{dx}$ for $y = \ln(3x^2 + 5)$.