- 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.
