## DBsubject('Calculus') ## DBchapter('Differentiation in Several Variables') ## DBsection('Partial Derivatives') ## KEYWORDS('calculus') ## TitleText1('Calculus: Early Transcendentals') ## EditionText1('2') ## AuthorText1('Rogawski') ## Section1('14.3') ## Problem1('3') ## Author('JustAsk - Vladimir Finkelshtein') ## Institution('W.H.Freeman') DOCUMENT(); loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); loadMacros("Parser.pl"); loadMacros("freemanMacros.pl"); loadMacros("PGauxiliaryFunctions.pl"); loadMacros("PGgraphmacros.pl"); loadMacros("PGchoicemacros.pl"); TEXT(beginproblem()); Context()->texStrings; $a=non_zero_random(-5,5,1);$a1=$a; if ($a==1) {$a1=''}; if ($a==-1) {$a1='-'};$ddy='\frac{\partial}{\partial{y}}'; $context = Context();$context->variables->add(y=>'Real'); $ans=Formula("$a*x/(y+$a*x)^2")->reduce(); BEGIN_TEXT \{ textbook_ref_exact("Rogawski ET 2e", "14.3","3") \}$PAR Use the Quotient Rule to compute: $PAR $$ddy\frac {y}{y+a1 x}=$$ \{ans_rule()\}$BR END_TEXT Context()->normalStrings; ANS($ans->cmp); Context()->texStrings; SOLUTION(EV3(<<'END_SOLUTION'));$PAR $SOL$BR We use the Quotient Rule to obtain: $PAR $$ddy \frac {y}{y+a1 x}=\frac{(y+a1 x)ddy (y)-y ddy (y+a1 x)}{\left(y+a1 x\right)^2}= \frac{(y+a1 x)\cdot 1 - y \cdot 1}{\left(y+a1 x\right)^2}=ans$$$BR END_SOLUTION ENDDOCUMENT();