%% %% 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 \( Right paren \) %% 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{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'.