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+%%
+%% This is file `easy.tex',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% probsoln.dtx  (with options: `easy.tex,package')
+%% Copyright (C) 2006 Nicola Talbot, all rights reserved.
+%% If you modify this file, you must change its name first.
+%% You are NOT ALLOWED to distribute this file alone. You are NOT
+%% ALLOWED to take money for the distribution or use of either this
+%% file or a changed version, except for a nominal charge for copying
+%% etc.
+%% \CharacterTable
+%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%%   Digits        \0\1\2\3\4\5\6\7\8\9
+%%   Exclamation   \!     Double quote  \"     Hash (number) \#
+%%   Dollar        \$     Percent       \%     Ampersand     \&
+%%   Acute accent  \'     Left paren    \(     Right paren   \)
+%%   Asterisk      \*     Plus          \+     Comma         \,
+%%   Minus         \-     Point         \.     Solidus       \/
+%%   Colon         \:     Semicolon     \;     Less than     \<
+%%   Equals        \=     Greater than  \>     Question mark \?
+%%   Commercial at \@     Left bracket  \[     Backslash     \\
+%%   Right bracket \]     Circumflex    \^     Underscore    \_
+%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
+%%   Right brace   \}     Tilde         \~}
+
+\newproblem{diffeasy:gpowh}{%
+\(f(x) = g(x)^{h(x)}.\)}{%
+\begin{eqnarray*}
+f(x) & = & e^{\ln g(x)^{h(x)}}\\
+ & = & e^{h(x)\ln g(x)}\\
+f'(x) & = & e^{h(x)\ln g(x)}(h'(x)\ln g(x) + h(x)\frac{g'(x)}{g(x)})\\
+ & = & g(x)^{h(x)}(h'(x)\ln g(x) + \frac{h(x)g'(x)}{g(x)})
+\end{eqnarray*}}
+
+\newproblem{diffeasy:arcsin}{%
+\(y = \arcsin(x)\)}{%
+\[\sin(y) = x\]
+diff. w.r.t. $x$:
+\begin{eqnarray*}
+\cos y \frac{dy}{dx} & = & 1\\
+\frac{dy}{dx} & = & \frac{1}{\cos y}\\
+ & = & \frac{1}{\sqrt{1 - \sin^2y}}\\
+ & = & \frac{1}{\sqrt{1-x^2}}.
+\end{eqnarray*}}
+
+\newproblem{diffeasy:arccos}{%
+$y = \arccos x$.}{%
+\(\cos y = x\)
+diff. w.r.t. $x$:
+\begin{eqnarray*}
+-\sin y \frac{dy}{dx} & = & 1\\
+\frac{dy}{dx} & = & \frac{-1}{\sin y}\\
+ & = & \frac{-1}{\sqrt{1-\cos^2y}}\\
+ & = & \frac{-1}{\sqrt{1-x^2}}
+\end{eqnarray*}}
+
+\newproblem{diffeasy:tan}{%
+\(y = \tan x\)}{%
+\begin{eqnarray*}
+y & = & \tan x\\
+ & = & \frac{\sin x}{\cos x}\\
+\frac{dy}{dx} & = & \frac{\cos x}{\cos x} + \sin x\times\frac{-1}{\cos^2x}\times -\sin x\\
+ & = & 1 + \tan^2x\\
+ & = & \sec^2x.
+\end{eqnarray*}}
+
+\newproblem{diffeasy:arctan}{%
+\(y = \arctan x = \tan^{-1}x\)}{%
+\[\tan y = x\]
+diff w.r.t. $x$:
+\begin{eqnarray*}
+\sec^2y\frac{dy}{dx} & = & 1\\
+\frac{dy}{dx} & = & \frac{1}{\sec^2y}\\
+ & = & \frac{1}{1+\tan^2y}\\
+ & = & \frac{1}{1+x^2}
+\end{eqnarray*}}
+
+\newproblem{diffeasy:cot}{%
+\(y = (\tan x)^{-1} = \cot x\)}{%
+\begin{eqnarray*}
+\frac{dy}{dx} & = & -(\tan x)^{-2}\sec^2x\\
+ & = & -\frac{\cos^2x}{\sin^2x}\cdot\frac{1}{\cos^2x}\\
+ & = & \frac{-1}{\sin^2x}\\
+ & = & -\csc^2x.
+\end{eqnarray*}}
+
+\newproblem{diffeasy:cosxsqsinx}{%
+$y = \cos(x^2)\sin x$.}{%
+\[\frac{dy}{dx} = -\sin(x^2)2x\sin x + \cos(x^2)\cos x\]}
+
+\newproblem{diffeasy:xlnx}{%
+$y = (x+1)\ln(x+1)$.}{%
+\begin{eqnarray*}
+\frac{dy}{dx} & = & \ln(x+1) + \frac{x+1}{x+1}\\
+ & = & 1 + \ln(x+1).
+\end{eqnarray*}}
+
+\newproblem{diffeasy:glng}{%
+$f(x) = g(x)\ln(g(x))$.}{%
+\begin{eqnarray*}
+f'(x) & = & g'(x)\ln(g(x)) + \frac{g(x)}{g(x)}g'(x)\\
+ & = & g'(x)(1+\ln(g(x))).
+\end{eqnarray*}}
+
+\newproblem{diffeasy:sinx/x}{%
+$y = \frac{\sin x}{x}$.}{%
+\[\frac{dy}{dx} = \frac{\cos x}{x} - \frac{\sin x}{x^2}\]}
+
+\endinput
+%%
+%% End of file `easy.tex'.