## DESCRIPTION
## This is Problem 2.4.24 from the APEX Calculus text. It covers differentiation of basic trig functions and the Quotient Rule.
## ENDDESCRIPTION

## DBsubject(Calculus - single variable)
## DBchapter(Differentiation)
## DBsection(Quotient rule (with trigonometric functions))
## Level(2)
## Institution('Valdosta State University')
## Author('S. V. Ault')
## RevisedBy('F. J. Francis')
## TitleText1('APEX Calculus')
## AuthorText1('Hartman')
## EditionText1('3.0')
## Section1('2.4')
## Problem1('24')
## MO(1)
## Keywords('derivative', 'quotient rule', 'trig functions', 'ULETH-MATH1560', 'ULETH-MATH1565')

###########################
#  Initialization
DOCUMENT();

# Load whatever macros you need for the problem
loadMacros(
  # REQUIRED: Used for basic question and answer operations.
  "PGstandard.pl",
  # REQUIRED: Used for expression parsing.
  "MathObjects.pl",
  # Usually required for proper text formatting.
  "PGML.pl",
  # Used to provide contextual help for how to type inequalities/intervals.
  "AnswerFormatHelp.pl"
);

# Sets up basic problem information.
TEXT(beginproblem());


#############################
#   Setup
#-ULETH-#
Context("Numeric")->noreduce('(-x)-y','(-x)+y');;

$a = random(2,10,1)*random(-1,1,2);
$b = random(1,10,1)*random(-1,1,2);
$c = random(1,10,1)*random(-1,1,2);

$f = Formula("$a*sin(x)")->reduce();
$df = $f->D('x');

$g = Formula("$b*cos(x)+$c")->reduce();
$dgu = $g->D('x');
$dg = Compute("$dgu")->reduce();


$func = Formula($f/$g)->reduce();
$ans = $func->D('x');
#-ENDULETH-#

#############################
#   Problem Text
#-ULETH-#

BEGIN_PGML
Compute the derivative of the given function.

>>[``f(x) = \frac{[$f]}{[$g]}.``]<<

[`f'(x) = `] [___________________________________________________] [@ AnswerFormatHelp("formulas") @]*
END_PGML
#-ENDULETH-#

#############################
#  Solution
#-ULETH-#

BEGIN_PGML_SOLUTION

You may compute this derivative using the Quotient Rule.

[``
  \begin{array}{rcl}
    f'(x) &=& \displaystyle
    \frac{ ([$g])\frac{d}{dx}[ [$f] ] - ([$f]) \cdot \frac{d}{dx}[ [$g] ]
    }{([$g])^2}\\
    &&\\
    &=& \displaystyle \frac{ ([$g])([$df]) - ([$f])([$dg]) }{ ([$g])^2 }.
  \end{array}
``]
END_PGML_SOLUTION
#-ENDULETH-#

#############################
#  Answer evaluation

#-ULETH-#
# Setting this to 1 means that students will receive feedback on whether their
# answers are correct.
$showPartialCorrectAnswers = 0;
#-ENDULETH#
ANS( $ans->cmp() );


#############################
#  Comments
    COMMENT('
    Randomization provides 7200 different possible versions of this question.<BR>
    Includes a solution set.<BR>
    Made from a ULETH template.<BR>
    ');
ENDDOCUMENT();