- WE WANT TO DETERMINE THE DERIVATIVE OF THE GIVEN FUNCTION THAT'S A QUOTIENT OF AN EXPONENTIAL FUNCTION WITH BASE E AND A POLYNOMIAL FUNCTION. SO BECAUSE WE HAVE A QUOTIENT WE'LL APPLY THE QUOTIENT RULE, WHICH IS GIVEN HERE JUST IN CASE YOU NEED TO REVIEW. SO TO APPLY THIS FORMULA HERE WE'LL LET THE NUMERATOR EQUAL F AND THE DENOMINATOR EQUAL G. SO TO DETERMINE F PRIME OF X WE'LL START WITH THE DENOMINATOR. NOTICE THE DENOMINATOR IS JUST A DENOMINATOR SQUARED, SO WE'LL HAVE THE QUANTITY X TO THE 3RD + 2 SQUARED. AND THE NUMERATOR'S GOING TO BE G x F PRIME - F x G PRIME. WELL, THE DENOMINATOR X TO THE 3RD + 2 x THE DERIVATIVE OF THE NUMERATOR. BE CAREFUL, NOTICE THE MINUS 1 IS PART OF THE EXPONENT. - THE NUMERATOR x THE DERIVATIVE OF THE DENOMINATOR. SO ONCE WE DETERMINE THESE TWO DERIVATIVES THE REST WILL BE ALGEBRA. NOMINATOR'S GOING TO STAY THE SAME. HERE WE'LL HAVE THE QUANTITY X TO THE 3RD + 2 x THE DERIVATIVE OF E TO THE POWER OF 2X SQUARED - 1. THIS IS GOING TO REQUIRE THE CHAIN RULE. SO U IS GOING TO BE THE EXPONENT. SO THE DERIVATIVE OF E TO THE U IS E TO THE U x U PRIME. SO WE'LL HAVE E TO THE POWER OF 2X SQUARED - 1 x THE DERIVATIVE OF 2X SQUARED - 1 THAT'LL BE 4X. - E TO THE POWER OF 2X TO THE 2ND - 1 x THE DERIVATIVE OF X CUBED + 2 WHICH IS 3X SQUARED. NOW LET'S DETERMINE THIS PRODUCT HERE AND THIS PRODUCT HERE. SO TO CLEAR THESE PARENTHESIS WE'LL HAVE TO DISTRIBUTE THIS PRODUCT HERE. SO WE'LL HAVE E TO THE POWER OF 2X SQUARED - 1 x 4X x X CUBED, SO WE'LL HAVE 4X TO THE 4TH E TO THE POWER OF 2X SQUARED - 1. AND THEN WE'LL HAVE + E TO THE POWER OF 2X SQUARED - 1 x 4X x 2 THAT'LL BE 8X E TO THE 2X SQUARED - 1. MINUS--HERE WE'LL HAVE 3X SQUARED E TO THE 2X SQUARED - 1. SO THE LAST STEP IN THIS PROBLEM WILL BE TO FACTOR THE NUMERATOR. LET'S GO AHEAD AND FINISH IT ON THE NEXT SLIDE. NOW WE WANT TO IDENTIFY THE GREATEST COMMON FACTOR OF THE NUMERATOR, WHICH IS GOING TO BE-- WHICH WILL BE ONE FACTOR OF X AND E TO THE POWER OF 2X SQUARED - 1. SO WHEN YOU FACTOR OUT THE GREATEST COMMON FACTOR OF X E TO THE POWER OF 2X SQUARED - 1 WE'RE LEFT WITH 4X CUBED + 8 - 3X. WELL, LET'S PUT THE - 3X SECOND AND THE + 8 THIRD SO THE TERMS ARE IN DESCENDING ORDER. SO HERE'S THE DERIVATIVE OF THE GIVEN FUNCTION. WE'LL TAKE A LOOK AT ONE MORE FUNCTION INVOLVING THE EXPONENTIAL FUNCTION WITH BASE E AND THE QUOTIENT RULE IN THE NEXT VIDEO.