A-Level Pure Mathematics — Differentiation practice problems

\documentclass[11pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage[margin=2cm]{geometry}
\usepackage{amsmath, amssymb}
\usepackage{enumitem}

\title{A-Level Pure Mathematics --- Differentiation Practice}
\author{}
\date{}

\begin{document}
\maketitle

\section*{Question 1 --- Standard derivatives}

Differentiate the following functions with respect to $x$:
\begin{enumerate}[label=\textbf{(\alph*)}]
  \item $f(x) = 5x^4 - 3x^2 + 7x - 2$
  \item $g(x) = (2x + 1)^3$ \hfill \textit{(use the chain rule)}
  \item $h(x) = x^2 \sin x$ \hfill \textit{(use the product rule)}
  \item $\displaystyle p(x) = \frac{\ln x}{x}$ \hfill \textit{(use the quotient rule)}
\end{enumerate}

\section*{Question 2 --- Stationary points}

Consider $f(x) = x^3 - 6x^2 + 9x + 1$.

\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Find $f'(x)$ and $f''(x)$.
  \item Find the coordinates of the stationary points of $y = f(x)$.
  \item Use the second derivative test to classify each stationary point as a local maximum, local minimum or point of inflection.
  \item Sketch the curve $y = f(x)$ for $-1 \leq x \leq 5$, showing clearly the stationary points and the $y$-intercept.
\end{enumerate}

\section*{Question 3 --- Optimisation}

A rectangular box with a square base of side $x$ cm and height $h$ cm has a fixed volume of $500$\,cm$^3$.

\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Show that the total surface area $S$ of the box (with no lid) is given by
  \[ S(x) = x^2 + \frac{2000}{x}. \]
  \item Find $\dfrac{dS}{dx}$ and the value of $x$ that minimises $S$.
  \item Confirm using the second derivative that this value gives a minimum.
  \item State the minimum surface area, giving your answer to the nearest whole cm$^2$.
\end{enumerate}

\section*{Question 4 --- Implicit differentiation}

A curve is defined implicitly by $x^2 + xy + y^2 = 7$.

\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.
  \item Find the equation of the tangent to the curve at the point $(1, 2)$.
\end{enumerate}

\end{document}