1 MATH 16A LECTURE. OCTOBER 21, 2008. PROFESSOR: WELCOME BACK. IS EVERYBODY READY TO GET GOING HERE? HELLO. LET'S SEE IF I CAN MAKE THE MICROPHONE WORK BETTER. HOW IS THAT? OKAY. SO I WANT TO KEEP GOING WITH WAYS TO DO DIFFERENTIATION AND APPLICATIONS TO NEW PROBLEMS THAT WE COULDN'T DO BEFORE. SO LET ME PUT UP A QUICK REVIEW OF THE THREE RULES WE HAD LAST TIME. JUST SO I CAN REFER TO THEM. WE HAD THE PRODUCT RULE. DIFFERENTIATING A PRODUCT OF TWO FUNCTIONS. AND HERE IT IS. (ON BOARD). AND THEN WE HAD THE QUOTIENT RULE. F-OF X-OVER G-OF X. AND IT LOOKS LIKE THIS. OKAY. SO THE ONLY DIFFERENCE BETWEEN THIS AND THIS IT NUMERATOR IS THAT SIGN RIGHT THERE. OTHERWISE THEY LOOK ALIKE. SO NOTICE THAT DIFFERENT IN THERE. AND THEN WE HAD THE CHAIN RULE. WHICH SAID IF YOU HAD TWO FUNCTIONS, FIRST YOU COMPUTED G-OF X-AND THEN YOU COMPUTED THAT, AND YOU WANT TO COMPUTE THE DERIVATIVE OF THE WHOLE FUNCTION CONSIDERED AS ONE FUNCTION, THEN FIRST YOU DIFFERENTIATE F. YOU GET F-PRIME AND MULTIPLY BY THE DERIVATIVE OF G. SO THOSE WERE THE THREE RULES SUMMARIZED IN THE WHOLE LAST LECTURE AND I'M GOING TO KEEP USING THEM TODAY. STUDENT: JUST ONE THING I WANT MAYBE WE WENT OVER LAST TIME WHAT IS THE DIFFERENCE BETWEEN THE CHAIN RULE AND GENERALIZE POWER RULE. PROFESSOR: GENERALIZED POWERFUL RULE WAS F-OF X-WAS P-TO A POWER AM NOW F-CAN BE ANY FUNCTION AT ALL THAT YOU CAN DIFFERENTIATE. AND THE GENERAL USED POWER RULE WAS INDEED A SPECIAL CASE OF THIS 2 FOR THE SPECIAL CASE WHEN F-WAS TAKE WHATEVER IT IS TO A POWER. SO WE TALKED, THE BOOK CHOOSE IT TEACH YOU THE GENERALIZED POWERRULE FIRST AND THEN LATER THIS GENERALIZATION. BUT NO IF YOU WANT YOU ONLY HAVE TO REMEMBER THE GENERALIZATION. EVERYTHING ELSE IS SPECIAL CASE. OKAY. SO LET ME NOW RESTATE THE CHAIN RULE USING SLIGHTLY DIFFERENT NOTATIONS SO WE CAN APPLY IF SOME CASE IS THAT MIGHT HAVE CONFUSING IN THE OLD NOTATION. SO LET ME RESTATE CHAIN RULE USING DIFFERENT NOTATION. SO IT'S GOING TO BE THE SAME MATHEMATICAL IDEAS. GOING TO COMBINE TWO FUNCTIONS, F-AND G-. SO U-IS G-OF X-AND THEN I PLUG THAT IN F. AND I GET Y. SO IF I PUT THOSE TWO TOGETHER I GET Y-EQUALS F-OF G-OF X. SO IT'S THE SAME FUNCTION AS BEFORE THAT I WANT TO DIFFERENTIATE. THAT SAME COMPOSITE FUNCTION. AND SO WHAT, SO NOW I CAN THINK OF THIS, WITH OR WITHOUT THAT VARIABLE U-IN THE MIDDLE. SOMETIMES WHEN THE PROBLEM IS STATED TO YOU THAT INTRODUCE A NEW VARIABLE. SO YOU HAVE TO JUST RECOGNIZE THAT. SO LET ME JUST DO IT IT TWO DIFFERENT WAYS HERE. SO I CAN THINK OF Y-AS A FUNCTION OF U-BECAUSE Y-EQUALS F-OF U-OR AS A FUNCTION OF X-BECAUSE Y-EQUALS F-OF G-OF X. IT'S THE SAME FUNCTION. JUST DIFFERENT NOTATION. SO LET ME WRITE DOWN THE DERIVATIVE IN TWO DIFFERENT WAYS, USE THE CHAIN RULE. SO I'M GOING TO WRITE IT DOWN USING THE OLD CHAIN RULE. THAT'S THE OLD CHAIN RULE. BUT NOW, G-OF X-EQUALS U. SO LET ME WRITE THAT DOWN. SO I JUST SUBSTITUTED G-OF X-EQUALS YOU CAN BECAUSE G-G-OF X-EQUALS U. AND NOW, LET ME DO ONE MORE SUBSTITUTION F-PRIME OF U, F-PRIME OF 3 U-I-SIMPLY WRITE D-Y-D-U. SO BECAUSE THAT'S WHAT THIS IMPLIES. THAT IMPLIES THAT D-Y-D-U-EQUALS F-PRIME OF U. SO JUST SUBSTITUTE THAT IN. AND THEN I HAVE G-PRIME OF X, WHAT DOES THIS IMPLY? BY DIFFERENTIATE THIS, U, D-U-D-X-EQUALS D-PRIME OF X. IT'S THE SAME OLD CHAIN RULE JUST USING DIFFERENT NAME. IF I PUT IT ALL TOGETHER AND 11 OUT ALL THE INTERMEDIATE STUFF, I GET THAT D-Y-D-X-IS D-Y-D-U-TIMES D-U-D-X. THAT'S THE CHAIN RULE AGAIN,NO NEW IDEAS. JUST IN A SLIGHTLY DIFFERENT NOTATION. D-Y-D-X-IS D-Y-D-U-D-Y-D-D-X-. NOW THIS IS SOMETIMES EASIER TO REMEMBER. DEPENDS ON YOU. SO WHY, IT MAYBE EASIER TO REMEMBER, IT KINDS OF LOOKS LIKE D-U-'S CANCEL. THEY DON'T CANCEL BECAUSE THIS IS A FUNCTION BUT IT KIND OF LOOKS LIKE D-U-'S CANCEL. AND WHAT DO YOU GET? YOU GET D-Y-D-X. SO IT LOOKS LIKE THE D-U-S CANCEL . OF COURSE, THEY REALLY DON'T. BUT IT HELPS YOU TO REMEMBER WHEN YOU'VE WRITTEN DOWN THE FORMULA RIGHT BECAUSE THEY CANCEL. STUDENT: U IS -- PROFESSOR: U-IS JUST THE NAME I'VE GIVEN TO THIS G-OF X. I'VE JUST NAME THE THING YOU GET IN THE MIDDLE. FIRST YOU HAVE TO COMPUTE G-OF X. CALL IT U. AND THEN PLUG THAT INTO F-TO GET YOUR FUNCTION. SO IT'S JUST THE CHAIN RULE WRITTEN DOWN IN OTHER NOTATION THAT YOU WILL FIND USED BY LOTS OF PEOPLE. NOT JUST THE BOOK. SO IT'S THE SAME OLD CHAIN RULE. AND NOW USE IT. I DIDN'T WRITE IT DOWN FOR FUN. I'M GOING TO USE IT. BUT ARE THERE ANY QUESTIONS? 4 OKAY. SO LET ME WRITE IT DOWN THIS WAY. SO FIND D-Y-D-U-IF Y-IS U-TO THE FIFTH MINUS TWO U-CUBED PLUS EIGHT. THAT'S A NICE SIMPLE POLYNOMIAL. AND U-IS X-SQUARED PLUS ONE. SO THERE IS U. AS FUNCTION OF X. SO THIS IF YOU LIKE IS MY G-OF X. AND HERE, IN THE NOTATION OVER THERE, Y-IS A FUNCTION OF U-SO THAT'S MY F-OF U. BUT I WON'T EVENLY BOTH. I DIDN'T NEED TO WRITE THIS DOWN. SO FIND D-Y-D-X. I WANT TO FIND OF DRIVEN OF Y. Y-DEPENDS ON U-AND U-DEPENDS ON X-. SO I WANT D-Y-D-X. I'M GOING TO USE THIS FORMULA D-Y-D-X-THAT COMES FROM OVER HERE. (ON BOARD). SO LET LET ME WRITE IT DOWN