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\lhead{Proof by Induction - Derivatives}
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\rhead{\href{https://danieljsmith.org}{danieljsmith.org}}
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\begin{document}
\vspace*{20pt}
\tableofcontents
\newpage
\begin{enumerate}[
  label=\textbf{\arabic*.},
  leftmargin=*,
  topsep=0pt,
  partopsep=0pt,
  parsep=0pt
]


% ---- Question 1 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Derivatives, Q1]

It is given that \[y = x^2 e^{ax}\] where $a$ is a constant. \\

Prove by mathematical induction that, for all integers $n \ge 2$,
\[
\frac{\mathrm{d}^n y}{\mathrm{d}x^n} = a^{n-2}\left(a^2x^2 + 2nax + n(n-1)\right)e^{ax}
\] \marks[-30pt]{6}
\newpage

% ---- Question 2 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Derivatives, Q2]

Prove by mathematical induction that, for every positive integer $n$,
\[
\frac{\mathrm{d}^n}{\mathrm{d}x^n}(x^3 e^{-x}) = (-1)^n e^{-x}\bigl(x^3 - 3n x^2 + 3n(n-1)x - n(n-1)(n-2)\bigr)
\] \marks[-30pt]{7}

\newpage


% ---- Question 3 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Derivatives, Q3]

Given that
\[
y=e^{-2x}\cosh 3x
\]
prove by induction that for $n \in \mathbb{N}$
\[
\frac{\mathrm{d}^n y}{\mathrm{d}x^n}=e^{-2x}\left(\frac{1+(-5)^n}{2}\cosh 3x+\frac{1-(-5)^n}{2}\sinh 3x\right)
\] \marks[-30pt]{6}

\newpage

% ---- Question 4 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Derivatives, Q4]

Let \[y = x^2\cosh x\]

Prove by induction that, for all integers $n \ge 1$,
\[
\frac{\mathrm{d}^{2n}y}{\mathrm{d}x^{2n}} = x^2\cosh x + 4nx\sinh x + 2n(2n-1)\cosh x
\] \marks[-20pt]{6}


\newpage

% ---- Question 5 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Derivatives, Q5]

A function is defined by $y=f(t)$ where $f(t)=(1+at)\ln(1+at)$ and $a$ is a constant. \\

\begin{enumerate}[label=\textbf{(\alph*)}]
  \item By considering $\frac{\mathrm{d}^2y}{\mathrm{d}t^2}$, $\frac{\mathrm{d}^3y}{\mathrm{d}t^3}$, $\frac{\mathrm{d}^4y}{\mathrm{d}t^4}$ and $\frac{\mathrm{d}^5y}{\mathrm{d}t^5}$, make a conjecture for a general formula for $\frac{\mathrm{d}^n y}{\mathrm{d}t^n}$ in terms of $n$, $a$ and $t$ for any integer $n \ge 2$. \marks{3} \\
  \item Use induction to prove the formula conjectured in part $(a)$. \marks{4} \\
  \item In the case when $f(t)=(1+3t)\ln(1+3t)$, find the rate at which the $6^{\mathrm{th}}$ derivative of $f(t)$ is varying when $t=\frac{5}{3}$. \marks{2} \\
\end{enumerate}

\newpage

% ---- Question 6 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Derivatives, Q6]

The function $f(x)$ is defined by $f(x)=(1+x)^{-\frac12}$, for $x>-1$. \\

\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Prove by mathematical induction that the $n$th derivative of $f(x)$, $f^{(n)}(x)$, for all $n \ge 1$, is given by
  \[
  f^{(n)}(x)=\frac{(-1)^n(2n)!}{2^{2n}n!(1+x)^{n+\frac12}}
  \]
  \marks[-20pt]{4}
  \item Hence prove that
  \[
  (1+x)^{-\frac12}=1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\ldots+(-1)^n\frac{(2n)!}{2^{2n}(n!)^2}x^n+\ldots
  \]
  for $-1<x\le 1$. \marks{3} \\
\end{enumerate}


\newpage

% ---- Question 7 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Derivatives, Q7]

It is given that \[y = (ax + 1)^2 \ln(ax + 1)\] where $a$ is a positive constant. Prove by mathematical induction that, for every integer $n \ge 3$,
\[
\frac{\mathrm{d}^n y}{\mathrm{d}x^n} = 2(-1)^{n-3} \frac{(n-3)!a^n}{(ax + 1)^{n-2}}
\] \marks[-20pt]{6}

\newpage

% ---- Question 8 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Derivatives, Q8]

The function $f$ is such that $f''(x)=-f(x)$. \\

Prove by mathematical induction that, for every positive integer $n$,
\[
\frac{\mathrm{d}^{2n}}{\mathrm{d}x^{2n}}\left(x^2f(x)\right)=(-1)^n\left(x^2f(x)-4nxf'(x)-2n(2n-1)f(x)\right)
\] \marks[-30pt]{7}


\end{enumerate}
\end{document}
