- IF F OF X = THE QUANTITY X SQUARED + 4X - 3
DIVIDED BY SQUARE ROOT X WE WANT TO FIND F PRIME OF X,
THE DERIVATIVE FUNCTION AND F PRIME OF 4.
NOTICE HOW THE GIVEN FUNCTION IS A QUOTIENT.
SO WE MIGHT BE THINKING THAT WE HAVE TO USE
THE QUOTIENT RULE GIVEN HERE BELOW IN RED
IN ORDER TO FIND OUR DERIVATIVE FUNCTION,
AND WE ACTUALLY CAN BUT BECAUSE WE'RE DIVIDING BY A SINGLE TERM
OR THE SQUARE ROOT X WE CAN ACTUALLY DIVIDE
EACH TERM IN THE NUMERATOR BY SQUARE ROOT X,
SIMPLIFY AND THEN FIND THE DERIVATIVE
USING THE BASIC POWER RULE INSTEAD.
SO FOR THIS VIDEO WE'RE GOING TO SHOW
HOW TO FIND THIS DERIVATIVE
WITHOUT USING THE QUOTIENT RULE
BUT I'LL PUT A LINK ON THE SCREEN TO ANOTHER VIDEO
WHERE WE'LL FIND THE SAME DERIVATIVE
USING THE QUOTIENT RULE.
YOU CAN COMPARE THE TWO METHODS.
BUT IT'S IMPORTANT TO EMPHASIS
THAT IF OUR DENOMINATOR WAS A SUM OR DIFFERENCE
WE WOULD BE REQUIRED TO USE THE QUOTIENT RULE
IN ORDER TO FIND THE DERIVATIVE.
BUT SINCE OUR DENOMINATOR IS A SINGLE TERM,
SQUARE ROOT X WE CAN AVOID THE QUOTIENT RULE.
WE ALSO NEED TO RECOGNIZE THAT WE CAN WRITE SQUARE ROOT X
AS X TO THE 1/2 BECAUSE THE EXPONENT ON THE X IS 1
AND THE INDEX IS 2.
WHICH MEANS WE'RE GOING TO REWRITE THIS
AND DIVIDE EACH TERM IN THE NUMERATOR BY X TO THE 1/2.
SO WE CAN WRITE F OF X AS X TO THE 2nd DIVIDED
BY X TO THE 1/2 + 4X DIVIDED BY X TO THE 1/2 - 3 DIVIDED BY X
TO THE 1/2 AND NOW WE'LL GO AND SIMPLIFY EACH FRACTION
AND WHEN DIVIDING AND THE BASES ARE THE SAME
WE SUBTRACT THE EXPONENTS.
SO WE CAN WRITE F OF X = X TO THE POWER OF 2 - 1/2
THAT WOULD BE 3/2 + 4 x X TO THE POWER OF 1 - 1/2
WHICH IS 1/2 AND THEN FOR THIS LAST TERM
TO ELIMINATE THE FRACTION WE'LL MOVE THIS UP TO THE NUMERATOR
WHICH WILL CHANGE THE SIGN OF THE EXPONENT.
SO WE'D HAVE - 3X TO THE POWER OF -1/2.
SO THIS FUNCTION HERE IS THE SAME
AS THE ORIGINAL FUNCTION BUT NOW WE DON'T HAVE A QUOTIENT.
SO NOW WE CAN FIND F PRIME OF X
BY APPLYING THE BASIC POWER RULE AND IF WE NEED A REVIEW
IT'S GIVEN HERE IN RED.
SO THE DERIVATIVE OF X TO THE 3/2 WE WOULD HAVE 3/2 x X
TO THE POWER OF 3/2 - 1 WHICH IS 3/2 - 2/2 OR 1/2 + 4
THEN WE'D HAVE x 1/2.
X TO THE POWER OF 1/2 - 1 IS -1/2 - 3 x -1/2.
X TO THE POWER OF -1/2 - 1 IS -3/2.
NOW LET'S GO AHEAD AND SIMPLIFY THIS.
THIS SHOULD BE + 2X TO THE - 1/2.
THIS WOULD BE + 3/2 X TO THE POWER OF -3/2.
NOW WE CAN LEAVE THE DERIVATIVE FUNCTION IN A VARIETY OF FORMS.
LET'S AT LEAST WRITE IT SO WE HAVE POSITIVE EXPONENTS.
SO F PRIME OF X = 3/2X TO THE 1/2 + 2/X TO THE 1/2
AND THEN WE'D HAVE + 3 DIVIDED BY 2X TO THE 3/2 POWER.
SO NOW IF WE WANT TO FIND F PRIME OF 4
WHICH WOULD GIVE US THE SLOPE OF THE TANGENT LINE AT X = 4
WE'LL SUBSTITUTE 4 FOR X.
NOW TO SAVE SOME TIME I'VE ALREADY DETERMINED THIS VALUE.
IT'S ACTUALLY EQUAL TO 67/16 WHICH IS EQUAL TO 4.1875.
REMEMBER THIS WOULD BE THE SLOPE OF THE TANGENT LINE AT X = 4.
SO AGAIN, HERE'S OUR DERIVATIVE FUNCTION
AND HERE'S F PRIME OF 4.
NOW REMEMBER IF YOU WANT TO YOU CAN SEE THIS SAME PROBLEM
USING THE QUOTIENT RULE IF YOU JUST FOLLOW THE LINK
ON THE SCREEN.
BUT BEFORE WE GO LET'S GO
AND TAKE A LOOK AT THE GRAPH OF OUR FUNCTION
AND THE SLOPE OF THE TANGENT LINE AT X = 4.
HERE'S THE GRAPH OF OUR FUNCTION
AND HERE'S THE POINT ON THE FUNCTION WHEN X = 4.
SO IF WE WERE TO SKETCH A TANGENT LINE AT THIS POINT
IT MIGHT LOOK SOMETHING LIKE THIS
AND SINCE F PRIME OF 4 WAS EQUAL TO 67/16
THIS IS ALSO THE SLOPE OF OUR TANGENT LINE.
I HOPE YOU FOUND THIS EXAMPLE HELPFUL.