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%%
%% This is file `implicit.tex',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% probsoln.dtx (with options: `implicit.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 \( Right paren \)
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%% Commercial at \@ Left bracket \[ Backslash \\
%% Right bracket \] Circumflex \^ Underscore \_
%% Grave accent \` Left brace \{ Vertical bar \|
%% Right brace \} Tilde \~}
\newproblem{imd:circ}{%
Find the gradient of the unit circle ($x^2 + y^2 = 1$).}{%
Differentiating with respect to $x$ gives:
\begin{eqnarray*}
2x + 2y\frac{dy}{dx} & = & 0\\
\frac{dy}{dx} & = & \frac{-2x}{2y}\\
& = & \frac{-x}{\sqrt{1-x^2}}.
\end{eqnarray*}}
\newproblem{imd:ysq:xcuov2mx}{%
Find $\frac{dy}{dx}$, given
\begin{displaymath}
y^2 = \frac{x^3}{2-x}
\end{displaymath}}{%
Differentiating both sides w.r.t.\ $x$:
\begin{eqnarray*}
2y\frac{dy}{dx} & = & \frac{(2-x)3x^2 - x^3(-1)}{(2-x)^2}\\
& = & \frac{3x^2(2-x) + x^3}{(2-x)^2}\\
& = & \frac{6x^2 - 3x^3 + x^3}{(2-x)^2}\\
& = & \frac{6x^2-2x^3}{(2-x)^2}\\
& = & 2x^2\frac{3-x}{(2-x)^2}
\end{eqnarray*}
Therefore
\begin{displaymath}
y\frac{dy}{dx} = x^2\frac{3-x}{(2-x)^2}
\end{displaymath}}
\newproblem{imd:exy:IIxay}{%
Differentiate w.r.t.\ $x$:
\begin{displaymath}
e^{xy} = 2x + y
\end{displaymath}}{%
Differentiating both sides w.r.t.\ $x$:
\begin{eqnarray*}
e^{xy}(1y + x\frac{dy}{dx}) & = & 2 + \frac{dy}{dx}\\
xe^{xy}\frac{dy}{dx} - \frac{dy}{dx} & = & 2 - ye^{xy} \\
\frac{dy}{dx}(xe^{xy}-1) & = & 2 - ye^{xy}\\
\frac{dy}{dx} & = & \frac{2-ye^{xy}}{xe^{xy}-1}
\end{eqnarray*}}
\endinput
%%
%% End of file `implicit.tex'.
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