- TO FIND THE DERIVATIVE OF THE GIVEN NATURAL LOG FUNCTION
IN IT'S CURRENT FORM,
WE'D HAVE TO APPLY THE CHAIN RULE
WHERE U, THE INNER FUNCTION,
WOULD BE THE SQUARE ROOT OF A QUOTIENT,
THEREFORE DETERMINING U PRIME WOULD BE PRETTY INVOLVED.
WHAT WE CAN DO IS APPLY THE PROPERTY OF LOGARITHMS
TO REWRITE THE FUNCTION
SO IT'S EASIER TO DETERMINE THE DERIVATIVE.
THE FIRST THING WE SHOULD RECOGNIZE IS THAT
WE CAN REWRITE THE SQUARE ROOT AS A RATIONAL EXPONENT.
THIS WOULD BE THE NATURAL LOG OF THE QUOTIENT
RAISED TO THE 1/2 POWER.
AND NOW IN THIS FORM,
WE CAN APPLY THE POWER PROPERTY OF LOGARITHMS STATED HERE
WHERE WE COULD TAKE THE EXPONENT AND MOVE IT TO THE FRONT
AND MULTIPLY IT BY THE NATURAL LOG.
SO WE CAN WRITE THE FUNCTION AS F OF X
= 1/2 x THE NATURAL LOG OF THE QUOTIENT.
AND NOW THAT WE HAVE THE LOG OF A QUOTIENT,
WE CAN REWRITE THIS IS A DIFFERENCE OF 2 LOGS
STATED BY THE QUOTIENT PROPERTY OF LOGARITHMS.
BUT WE'VE GOT TO BE A LITTLE BIT CAREFUL WHEN WE DO THIS.
WE WOULD HAVE F OF X = 1/2 x THE DIFFERENCE OF THE 2 LOGS
WHICH WOULD BE NATURAL LOG OF 3X TO THE 4th + 2 - NATURAL LOG 4X.
AND NOW, WE COULD DETERMINE THE DERIVATIVE IN THIS FORM,
BUT I THINK IT'S EASIER IF WE DISTRIBUTE THE 1/2 FIRST.
SO 1/2 NATURAL LOG 3X TO THE 4th + 2 - 1/2 NATURAL LOG 4X.
SO THE DERIVATIVE FUNCTION IS GOING TO BE EQUAL TO 1/2
x THE DERIVATIVE OF NATURAL LOG 3X TO THE 4th + 2, THIS IS A 4,
SO WE'LL HAVE TO APPLY THE CHAIN RULE AND HERE'S THE FORMULA.
SO WE'LL LET THE INNER FUNCTION = U,
SO HERE U WOULD BE EQUAL TO 3X TO THE 4th + 2.
SO THE DERIVATIVE OF NATURAL LOG U IS EQUAL TO 1/U x U PRIME,
SO HERE WE'D HAVE 1/3X TO THE 4th + 2
x THE DERIVATIVE OF 3X TO THE 4th + 2,
THAT WOULD BE 12X TO THE 3rd,
WE'LL PUT THAT /1, - 1/2 x THE DERIVATIVE OF NATURAL LOG 4X.
AGAIN, WE HAVE A COMPOSITE FUNCTION.
SO HERE, U WOULD BE EQUAL TO 4X.
SO THE DERIVATIVE IS 1/U x U PRIME OR 1/4X x 4.
NOW, LET'S GO AHEAD AND SIMPLIFY THESE PRODUCTS
AND SEE WHAT WE HAVE.
HERE WE HAVE A COMMON FACTOR OF 2.
THIS SIMPLIFIES TO 1, THIS WOULD BE 6,
AND OVER HERE THESE 4s SIMPLIFY OUT.
SO IT LOOKS LIKE WE HAVE 6X CUBED
/THE QUANTITY 3X TO THE 4th + 2 -, HERE WE'D HAVE 1/2X.
SO THIS IS OUR DERIVATIVE FUNCTION,
BUT NORMALLY WE'RE EXPECTED TO COMBINE THESE TWO FRACTIONS.
SO LET'S CONTINUE THIS ON THE NEXT SLIDE.
WELL, THE LEAST COMMON DENOMINATOR
OF THESE TWO FRACTIONS
WOULD BE THE PRODUCT OF THE TWO DENOMINATORS.
SO MULTIPLY THIS FIRST FRACTION BY 2X/2X.
WE'LL MULTIPLY THE SECOND FRACTION BY 3X TO THE 4th + 2
/3X TO THE 4th + 2.
SO NOTICE OUR COMMON DENOMINATOR WOULD BE 2X
x THE QUANTITY 3X TO THE 4th + 2,
AND NOW WE CAN COMBINE THE NUMERATORS.
HERE WE'D HAVE 12X TO THE 4th.
AND NOW, YOU NEED TO BE CAREFUL HERE
BECAUSE WE'RE SUBTRACTING THIS PRODUCT,
SO WE'D HAVE - 3X TO THE 4th AND THEN - 2.
SO WE HAVE 2 LIKE TERMS IN THE NUMERATOR,
SO OUR DERIVATIVE FUNCTION WOULD BE THE QUANTITY 9X TO THE 4th
MINUS 2 ALL/2X x THE QUANTITY 3X TO THE 4th + 2.
NOTHING FACTORS HERE AND THERE ARE NO COMMON FACTORS
BETWEEN THE NUMERATOR AND DENOMINATOR,
SO THIS WOULD BE OUR DERIVATIVE FUNCTION.
I HOPE YOU FOUND THIS HELPFUL.