##DESCRIPTION
##KEYWORDS('derivatives', 'trigonometry', 'product rule')
##  Find a derivative of a function involving trigonometric functions,
##  and evalute it at a given point; requires using product rule
##ENDDESCRIPTION

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$pi = arccos(-1);

$a_n = random(2,12,1);
$a_s = random(-1,1,2);
$a   = $a_n * $a_s;

$x_d = random(3,6,1);
while ($x_d == 5) {$x_d = random(3,6,1);};

$q = random(0,1,1);
if ($q == 0) {
    $x_n = 1;
    $x_s = -1;
    $x_sign = '-';
    };
if ($q == 1) {
    $x_n = 1;
    $x_s = 1;
    $x_sign = '';
    };
if ($q == 2) {
    $x_n = $x_d - 1;
    $x_s = 1;
    $x_sign = '';
    };
if ($q == 3) {
    $x_n = $x_d + 1;
    $x_s = 1;
    $x_sign = '';
    };
if ($q == 4) {
    $x_n = 2 * $x_d - 1;
    $x_s = 1;
    $x_sign = '';
    };

if ($x_n != 1) { $x_num = $x_n };
if ($x_n == 1) { $x_num = '' };

$x = $x_s*$x_n*$pi/$x_d;

$deriv1 =($a*(sin($x)+cos($x))+ $a*$x*(cos($x)-sin($x)));
$funct1 ="($a*(sin(x)+cos(x))+ $a*x*(cos(x)-sin(x)))";

TEXT(EV2(<<EOT));
Let \[ f(x) = $a x( \sin x + \cos x) \]
$PAR
\( f'( x ) = \)
$PAR
\{ans_rule(45) \}
$PAR
\( f'( $x_sign \frac { $x_num \pi } {$x_d} ) = \)
$PAR
\{ans_rule(45) \}

EOT

$ans = $funct1;
ANS(fun_cmp($ans));

$ans = $deriv1;
ANS(num_cmp($ans));

ENDDOCUMENT();        # This should be the last executable line in the problem.