- WE'RE GIVEN F OF X = 1 DIVIDED BY THE QUANTITY 3X - 2
RAISED TO THE 3rd.
WE WANT TO FIND F PRIME OF X, THE DERIVATIVE FUNCTION,
AND F PRIME OF 1.
THE FIRST THING WE SHOULD RECOGNIZE
IS THAT THE GIVEN FUNCTION IS A COMPOSITE FUNCTION,
WHICH MEANS WE'LL HAVE TO APPLY THE CHAIN RULE
TO FIND THE DERIVATIVE.
BUT WE WANT TO FIND THE EXTENDED POWER RULE GIVEN HERE,
WHICH IS THE BASIC POWER RULE WITH THE CHAIN RULE BUILT IN.
TO DO THIS THOUGH,
WE'LL HAVE TO CHANGE THE FORM OF THE FUNCTION.
WHERE IF WE HAVE F OF X = 1 DIVIDED BY THE QUANTITY 3X - 2
RAISED TO THE 3rd,
TO AVOID USING THE QUOTIENT RULE
WE CAN MOVE THIS UP TO THE NUMERATOR,
WHICH WOULD CHANGE THE SIGN OF THE EXPONENT.
THIS WOULD BE EQUIVALENT TO F OF X = THE QUANTITY 3X - 2
RAISED TO THE POWER OF -3.
IN THIS FORM WE CAN GO AHEAD
AND APPLY THE EXTENDED POWER RULE GIVEN HERE,
WHERE AGAIN, U IS THE INNER FUNCTION.
SO NOTICE IN THIS CASE U = 3X - 2.
SO IF WE HAVE U = 3X - 2, WE ALSO NEED U PRIME.
U PRIME WOULD JUST BE 3.
SO NOW WE CAN WRITE THE ORIGINAL FUNCTION F OF X
AS F OF X = U TO THE POWER OF -3.
AND NOW, TO FIND F PRIME OF X
WE'LL APPLY THE EXTENDED POWER RULE.
WE MULTIPLY BY THE EXPONENT.
THAT'S GOING TO BE -3 x U TO THE--
WE NEED TO BE CAREFUL HERE.
WE'RE SUBTRACTING 1 FROM THE EXPONENT.
-3 - 1 = -4, AND THEN x U PRIME.
WELL, WE HAVE U AND U PRIME RIGHT HERE,
WHICH MEANS F PRIME OF X = -3 x THE QUANTITY 3X - 2
RAISED TO THE POWER OF -4 x 3.
SO F PRIME OF X = -9 x THE QUANTITY 3X - 2
TO THE POWER OF -4.
SO THIS MAY BE FINE AS OUR DERIVATIVE FUNCTION.
LET'S ALSO SHOW HOW WE CAN WRITE THIS USING A POSITIVE EXPONENT.
WE CAN MOVE THE BASE OF 3X - 2 DOWN TO THE DENOMINATOR AGAIN,
WHICH WILL CHANGE THE SIGN OF THE EXPONENT.
SO WE COULD ALSO SAY F PRIME OF X = -9 IN THE NUMERATOR
DIVIDED BY THE QUANTITY 3X - 2 RAISED TO THE +4th POWER.
AND NOW FOR THE SECOND PART OF THE QUESTION,
WE WANT TO FIND F PRIME OF 1,
WHICH WOULD GIVE US THE SLOPE OF THE TANGENT LINE AT X = 1,
OR ALSO THE INSTANTANEOUS RATE OF CHANGE OF THE FUNCTION
AT X = 1.
SO F PRIME OF 1,
LET'S GO AHEAD AND USE THE FUNCTION IN FRACTION FORM.
IT WOULD BE -9 DIVIDED BY THE QUANTITY 3 x 1 - 2
RAISED TO THE 4th,
WHICH WOULD BE -9 DIVIDED BY-- WELL, 3 - 2 = 1.
WE HAVE 1 TO THE 4th, WHICH OF COURSE IS JUST 1.
THAT'S EQUAL TO -9.
SO F PRIME OF 1 = -9.
LET'S GO AHEAD AND VERIFY THIS GRAPHICALLY
BY LOOKING AT THE SLOPE OF THE TANGENT LINE AT X = 1.
HERE'S THE GRAPH OF OUR FUNCTION.
NOTICE WHEN X = 1 THE POINT ON THE FUNCTION,
OR POINT OF TANGENCY, IS THIS POINT HERE, THE .1,1.
TO FIND THIS Y COORDINATE,
WE'D HAVE TO SUBSTITUTE 1 FOR X
INTO THE DIVISIONAL FUNCTION GIVEN HERE.
AND BECAUSE F PRIME OF 1 = -9,
THE SLOPE OF THIS RED TANGENT LINE IS -9.
I HOPE YOU FOUND THIS HELPFUL.