Use the chain rule, if 𝑦 = 𝑓(𝑢) and 𝑢 = 𝑔(𝑥) then 𝑑𝑦 𝑑𝑥 = 𝑑𝑦 𝑑𝑢 × 𝑑𝑢 𝑑𝑥, to determine the derivative of composite functions involving power and polynomial functions.

Consider the function $f(x) = (4x^2 - 5x + 1)^3$. (a) State the function $u(x)$ where $y = [u(x)]^3$. [1] (b) Determine $\frac{du}{dx}$. [1] (c) Use the chain rule to determine $f'(x)$. [1]

A manufacturing company models the production cost per unit, \(C\), in dollars, using the function \(C(x) = 45(2x^2 - 5x + 8)^4\), where \(x\) represents the number of thousands of units produced. (a) Use the chain rule to determine \(\frac{dC}{dx}\). (2 marks) (b) Hence, determine the rate of change of cost per unit with respect to production level when \(x = 3\). (2 marks) (c) Determine \(\frac{d^2C}{dx^2}\) when \(x = 3\). (1 mark)

Consider the function $f(x) = (5x^2 - 3x + 2)^4$. (a) Use the chain rule to determine $f'(x)$. [2 marks] (b) Hence, evaluate $f'(1)$. [1 mark]

Determine the derivative of \( f(x) = (3x^2 + 2)^5 \).

Consider the function \( h(x) = (2x^3 - 5x + 7)^4 \). (a) Use the chain rule to determine \( h'(x) \). [2 marks] (b) Hence determine \( h''(x) \). [2 marks] (c) Evaluate \( h''(1) \), giving your answer correct to the nearest whole number. [1 mark]