Calculus January 25, 20041Political Science 552Instant CalculusWhat Is Differential Calculus?()3223 XXXgY ++==02040608010012014016001234XY13956332327130YXWhat Is a Slope of a Curve?XYslope∆∆=02040608010012014016001234XYCalculus January 25, 20042Getting the Slope3000223 XXY ++=Add ∆Y and ∆X:()()3000223 XXXXYY ∆++∆++=∆+Subtract first from second:()()()300300223223 XXXXXXY ++−∆++∆++=∆Getting the Slope-continued I() ()202036622 XXXXXXY ∆+∆+∆+∆=∆() ()XXXXXXXXY∆∆+∆+∆+∆=∆∆202036622XXXXXY∆++∆+=∆∆02026622Cancel ∆X on right side:Simplify:Getting the Slope-continued IIAs ∆X→0:262 XXY+=∆∆General transformation rule for:()1−××==ppXkpdXkXddXdYpkXY =Calculus January 25, 20043Example3223 XX ++Apply transformation rule:220162232130 XXXX +=×+×+×−310223 XXX ++Product Rule23232XXbXa++==Let:()54323246232 XXXXXXbaY ++=++=×=432101618 XXXdXdY++=Product Rule-continuedadXdbbdXdadXbaddXdY+=×=)(The Rule:()()()3322222236 XXXXXdXdY++++=432434321016184461218XXXXXXXXdXdY++=++++=Calculus January 25, 20044Square Ruleba =2aaaY =×=adXdaadXdaadXdadXaaddXdY2)(=+=×=The Rule:Partial DerivativesZXZXY 4232+++=11102004123−+++= ZXZZXZXYZXXY 44 +=∂∂241ZXZY+=∂∂Rewrite:Partial