## DESCRIPTION
## Calculus
## ENDDESCRIPTION

## KEYWORDS('derivative' 'implicit')
## Tagged by tda2d

## DBsubject('Calculus')
## DBchapter('Differentiation')
## DBsection('Implicit Differentiation')
## Date('')
## Author('')
## Institution('Union College')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')

DOCUMENT();

loadMacros(
   "PG.pl",
   "PGbasicmacros.pl",
   "PGchoicemacros.pl",
   "PGanswermacros.pl",
   "PGauxiliaryFunctions.pl",
   "PGunion.pl",        # Union College utilities
   "PGcourse.pl",       # Customization file for the course
);

TEXT(beginproblem());
BEGIN_PROBLEM();

$x0 = non_zero_random(-2,2,1);
$y0 = non_zero_random(-2,2,1);
$a = random(2,6,1);
$b = random(2,4,1);
while (3*$b*$y0**2 - 4*($a*$x0 - $y0)**3 == 0)
{
    $b = random(2,4,1);
}
$c = ($a*$x0-$y0)**4 + $b*$y0**3;

BEGIN_TEXT
For the equation given below, evaluate \(\displaystyle\frac{dy}{dx}\) 
at the point \(($x0,$y0)\):
\[
    ($a x - y)^4 + $b y^3 = $c.
\]
$PAR
\(\displaystyle\frac{dy}{dx}\) at \(($x0,$y0)\) = \{ans_rule(20) \}
END_TEXT

$showPartialCorrectAnswers = 1;
$ans = (-4*$a*($a*$x0-$y0)**3)/(3*$b*$y0**2 - 4*($a*$x0 - $y0)**3);
ANS(num_cmp($ans));

END_PROBLEM();
ENDDOCUMENT();