Understand the concept of the derivative as a function.

A function is defined as \( f(x) = 2x^3 - 6x + 1 \). Using the first principle of differentiation, determine \( f'(x) \) and evaluate \( f'(1) \).

A function is defined by $f(x) = 3x^2 - 4x + 1$ for $x \in \mathbb{R}$. Which of the following correctly describes the derivative function $f'(x)$?

Consider the function \( f(x) = x^3 - 6x^2 + 9x + 2 \). (a) Determine \( f'(x) \), the derivative function of \( f(x) \). [1 mark] (b) Evaluate \( f'(1) \) and \( f'(3) \). [1 mark] (c) Use the derivative function to determine the \( x \)-coordinates of all stationary points of \( f(x) \). [2 marks]

Given that \( h(x) = \frac{e^x}{x} \), determine the derivative function \( h'(x) \) in simplest form. Then evaluate \( h'(1) \).