- IN THIS VIDEO WE'LL TAKE 
A LOOK AT SOME EXAMPLES
OF DETERMINING THE DERIVATIVE 
OF EXPONENTIAL FUNCTIONS
WHEN THE BASE IS NOT EQUAL TO E.
OUR FORMULAS ARE GIVEN 
BELOW HERE IN RED.
IN MY LESSON VIDEO I DO EXPLAIN 
WHERE THESE FORMULAS COME FROM,
BUT THIS VIDEO FOCUSES 
ON PROVIDING
ADDITIONAL EXAMPLES.
NOTICE THE FIRST DERIVATIVE 
FORMULA IS FOR THE MOST BASIC
EXPONENTIAL FUNCTION.
THE DERIVATIVE OF "A" 
TO THE POWER OF X
WITH RESPECTS TO X IS EQUAL 
TO NATURAL LOG x "A" TO THE X.
BUT IF WE HAVE A COMPOSITE 
FUNCTION WHERE THE EXPONENT U
IS A FUNCTION OF X 
THE DERIVATIVE OF "A"
TO THE U WITH RESPECTS 
TO X IS EQUAL
TO NATURAL LOG "A" x "A" 
TO THE U x U PRIME.
SO NOTICE HOW EVEN IF YOU TRIED 
TO APPLY THE CHAIN RULE
TO "A" TO THE X WHERE U 
WOULD BE EQUAL TO X,
"A" PRIME WOULD JUST BE 1.
SO YOU'LL NEVER GO WRONG 
BY TRYING TO APPLY
THE CHAIN RULE.
LOOKING AT OUR FIRST EXAMPLE, 
WE HAVE F OF X = 5
TO THE POWER OF X.
SO NOTICE THAT OUR BASE OF 5 
IS GOING TO BE EQUAL TO "A".
AND SINCE OUR EXPONENT IS X, 
WE DON'T NEED TO APPLY
THE CHAIN RULE.
SO F PRIME OF X IS EQUAL 
TO NATURAL LOG "A"
OR NATURAL LOG 5 x "A" TO THE X, 
WHICH IS 5 TO THE POWER OF X.
ON OUR SECOND EXAMPLE NOTICE 
THE BASE IS 2,
SO THAT MEANS "A" IS EQUAL TO 2.
OUR EXPONENT 
IS THE FUNCTION OF X.
WE HAVE A COMPOSITE FUNCTION, 
SO WE'LL APPLY THE CHAIN RULE
WHERE THE INNER FUNCTION 
OR U IS EQUAL
TO X TO THE 3RD - 1.
SO WE'LL ALSO NEED U PRIME.
U PRIME IS GOING TO BE 
THE DERIVATIVE OF U
WITH RESPECTS TO X 
THAT'LL BE 3X SQUARED.
NOW WE HAVE ALL THE INFORMATION 
WE NEED, F PRIME OF X
IS GOING TO BE EQUAL 
TO A NATURAL LOG "A,"
WHICH IS NATURAL LOG 2 x A 
TO THE POWER OF U,
WHICH IS THE ORIGINAL FUNCTION.
2 TO THE POWER OF X CUBED
- 1 x U PRIME AND U PRIME
IS 3X SQUARED.
SO THIS IS OUR 
DERIVATIVE FUNCTION.
LET'S GO AHEAD AND CHANGE 
THE ORDER OF THIS PRODUCT.
F PRIME OF X IS GOING TO BE 
EQUAL TO 3X SQUARED
NATURAL LOG 2 x 2 
TO THE POWER OF X CUBED - 1.
WE'LL TAKE A LOOK AT SOME MORE 
EXAMPLES IN THE NEXT VIDEO.
