- WE WANT TO DETERMINE THE DERIVATIVE
OF THE GIVEN FUNCTION THAT INVOLVES
THE EXPONENTIAL FUNCTION WITH BASE E.
AS YOU CAN SEE, WE DO HAVE A QUOTIENT,
SO WE WILL BE APPLYING THE QUOTIENT RULE GIVEN HERE.
BUT I THINK IT MIGHT BE HELPFUL TO MANIPULATE THIS FUNCTION
BEFORE WE DETERMINE THE DERIVATIVE.
LOOKING AT THESE NEGATIVE EXPONENTS,
WE COULD REWRITE THESE AS FRACTIONS AS 1/E TO THE X.
LET'S GO AHEAD AND DO THAT.
SO TO WRITE THIS IN A DIFFERENT FORM
WE'D HAVE E TO THE X - 1/E TO THE X DIVIDED
BY E TO THE X + 1/E TO THE X.
NOW, OF COURSE WE DON'T WANT TO APPLY THE QUOTIENT RULE
WITH THESE FRACTIONS IN HERE.
BUT IF WE MULTIPLY THIS
BY E TO THE X/E TO THE X THESE FRACTIONS WILL SIMPLIFY OUT.
LET'S SEE WHAT THIS GIVES US.
F OF X IS GOING TO BE EQUAL TO-- WELL, E TO THE X x E TO THE X,
WHEN THE BASES ARE THE SAME WE ADD THE EXPONENTS.
WE'D HAVE E TO THE 2X - 1/E TO THE X x E TO THE X
WOULD BE 1.
THE DENOMINATOR WE WOULD HAVE E TO THE 2X AND THEN + 1.
SO I THINK IT'S GOING TO BE A LITTLE BIT EASIER
TO APPLY THE QUOTIENT RULE WHEN THE FUNCTION
IS IN THIS FORM.
SO FROM HERE WE'LL LET THE NUMERATOR EQUAL F
AND THE DENOMINATOR EQUAL G, AND APPLY THE QUOTIENT RULE.
SO TO DETERMINE THE DERIVATIVE OF THE GIVEN FUNCTION
WE'LL START WITH THE DENOMINATOR.
THE DENOMINATOR IS JUST THE NOMINATOR SQUARED,
SO WE'LL HAVE E TO THE POWER OF 2X + 1 SQUARED.
THE NUMERATOR'S GOING TO BE G x F PRIME - F x G PRIME.
WELL, THE DENOMINATOR
x THE DERIVATIVE OF THE NUMERATOR
- THE NUMERATOR
x THE DERIVATIVE OF THE DENOMINATOR.
NOW, FOR THE NEXT STEP WE'LL DETERMINE
THIS DERIVATIVE HERE AND THIS DERIVATIVE HERE.
SO LET'S CONTINUE THIS ON THE NEXT SLIDE.
NOW WE'LL DETERMINE THESE DERIVATIVES IN YELLOW.
TO DETERMINE THE DERIVATIVE OF E TO THE 2X,
THIS IS A COMPOSITE FUNCTION, SO U WOULD BE EQUAL TO 2X
AND THE DERIVATIVE OF E TO THE U WOULD BE E TO THE U x U PRIME
OR E TO THE 2X x 2.
WE'LL WRITE THAT AS 2 E TO THE 2X - QUANTITY E
TO THE 2X - 1 x THE DERIVATIVE OF E TO THE 2X + 1.
WELL, AGAIN, WE'RE GOING TO HAVE U = 2X,
SO THE DERIVATIVE WILL BE E TO THE 2X x 2 OR JUST 2 E
TO THE 2X.
SO NOW WE'LL DISTRIBUTE HERE AND THEN SIMPLIFY THE NUMERATOR.
SO WE'LL DISTRIBUTE HERE, HERE, HERE, AND HERE.
SO E TO THE 2X x 2 E TO THE 2X.
REMEMBER IF THE BASES ARE THE SAME
WE ADD THE EXPONENTS.
SO WE'LL HAVE 2 E TO THE 4X + 2 E TO THE 2X.
AND OVER HERE WE'LL HAVE - 2 E TO THE 4X.
AND THEN WE'LL HAVE - -2 E TO THE 2X THAT'LL BE + 2 E
TO THE 2X.
NOTICE HOW THESE TWO TERMS ARE OPPOSITES,
AND THESE TWO TERMS ARE LIKE TERMS.
SO OUR DERIVATIVE IS GOING TO BE EQUAL TO 4 E TO THE 2X
DIVIDED BY THE QUANTITY E TO THE 2X + 1 SQUARED.
SO HERE'S OUR DERIVATIVE FUNCTION.
SO GOING BACK TO THE BEGINNING JUST FOR A MOMENT,
WE DIDN'T HAVE TO MANIPULATE THIS FUNCTION TO DETERMINE
THE DERIVATIVE, BUT NOTICE HOW WHEN WE DID MANIPULATE
THIS TO ELIMINATE THE EXPONENTS OF -X.
IT DID RESULT IN A NICE CLEAN DERIVATIVE AS WE SEE HERE.
I HOPE YOU FOUND THIS EXAMPLE HELPFUL.