- WELCOME TO 
DIFFERENTIATION TECHNIQUES.
TODAY WE WILL BE COVERING 
THE CONSTANT RULE,
THE POWER RULE, 
THE CONSTANT MULTIPLE RULE,
AND THE SUM AND DIFFERENCE RULE.
THE GOAL OF THIS VIDEO WILL BE
TO FIND DERIVATIVES USING 
THE RULES MENTIONED ABOVE.
I WOULD LIKE TO START BY TAKING 
A LOOK AT THE GRAPH.
HERE WE HAVE THE GRAPH 
OF A CONSTANT FUNCTION,
AND WHAT WE'RE GOING TO DO
IS WE'RE GOING TO PLOT 
THE VALUES
OF THE DERIVATIVES 
AT VARIOUS POINTS.
SO WHEN WE START TO ANIMATE 
THIS POINT YOU CAN SEE IN RED
IT'S GRAPHING THE VALUE 
OF THE DERIVATIVE
AT THE CORRESPONDING X-VALUE.
REMEMBER THE VALUE 
OF THE DERIVATIVE
WOULD BE THE SLOPE 
OF THE TANGENT LINE
AT THAT GIVEN X-VALUE.
WHAT WE SEE HERE IS THAT THE 
DERIVATIVE IS ALWAYS EQUAL TO 0
AT ANY POINT 
ON A HORIZONTAL LINE,
AND HOPEFULLY THAT MAKES SENSE.
IF I WAS TO PICK A POINT 
ON THE HORIZONTAL LINE
AND SKETCH A TANGENT LINE,
OF COURSE IT WOULD BE THE SAME 
LINE WITH A SLOPE OF 0
WHICH BRINGS US 
TO OUR FIRST DERIVATIVE RULE,
THE CONSTANT RULE.
THE DERIVATIVE OF A CONSTANT 
IS 0, THAT IS,
IF C IS A REAL NUMBER,
CAN THE DERIVATIVE OF C WITH 
RESPECTS TO X STILL EQUAL 0.
A COUPLE QUICK EXAMPLES.
THE DERIVATIVE OF 5 WITH 
RESPECTS TO X WOULD BE 0.
THE DERIVATIVE OF PI WITH 
RESPECTS TO X WOULD BE 0.
REMEMBER PI IS ALSO A CONSTANT.
NEXT, WE'RE GOING TO LOOK 
AT THE POWER RULE,
BUT FIRST I WANT TO GO BACK 
AND TAKE A LOOK AT SOME GRAPHS
AND SEE IF WE CAN SEE A PATTERN
BETWEEN A FUNCTION 
AND ITS DERIVATIVES.
IN THIS CASE, 
WE'RE LOOKING AT A LINE.
REMEMBER A LINE 
IS A DEGREE 1 FUNCTION.
AS WE ANIMATE THIS POINT,
AGAIN THE DERIVATIVE WILL BE 
PLOTTED IN RED,
AND WE CAN SEE THE PATTERN HERE.
WE HAVE A DEGREE 1 FUNCTION, 
AND THE DERIVATIVE IS DEGREE 0.
IF WE LOOK AT A DEGREE 2 
FUNCTION OR A QUADRATIC FUNCTION
AND PLOT ITS DERIVATIVE IN RED,
WE CAN SEE THE DEGREE 2 FUNCTION 
HAS A DEGREE 1 DERIVATIVE.
IF I POSIT THIS POINT,
WE CAN SEE THE DERIVATIVE 
HAS A VALUE OF -2
WHICH WOULD REPRESENT THE SLOPE 
OF THIS TANGENT LINE.
LET'S GO AHEAD AND TAKE A LOOK 
AT A DEGREE 3 FUNCTION.
WE'LL ANIMATE THE POINT 
AND PLOT THE DERIVATIVE,
SO HERE'S A DEGREE 3 FUNCTION,
AND WE CAN SEE THAT 
THE DERIVATIVE
IS A QUADRATIC OR A DEGREE 2.
AGAIN, IF I WAS TO PAUSE 
THE TANGENT LINE,
LET'S SAY, RIGHT HERE AT X = -2,
THE VALUE OF THE RED FUNCTION, 
LOOKS LIKE IT WOULD BE -12,
WOULD TELL US THE SLOPE 
OF OUR TANGENT LINE.
SO WE KIND OF SEE 
A PATTERN HERE.
LET'S SUMMARIZE THE PATTERN 
THAT WE'VE JUST SEEN.
THE POWER RULE, 
IF N IS A RATIONAL NUMBER
THEN THE FUNCTION 
IS DIFFERENTIABLE,
AND THE DERIVATIVE OF X 
TO THE POWER OF N
WITH RESPECTS TO X = N 
x X TO THE POWER OF N - 1.
SO TO FIND THE DERIVATIVE
WE MULTIPLY 
BY OUR CURRENT EXPONENT,
AND THEN OUR NEW EXPONENT 
WE HAVE TO SUBTRACT 1.
WE SAW THE DEGREE 
OF THE FUNCTIONS WE GRAPHED
DECREASE BY 1 WHEN WE FOUND 
ITS DERIVATIVE.
THIS RULE DOES VERIFY 
THE PATTERNS THAT WE SAW.
NOW, OF COURSE, 
WE COULD PROVE THIS FORMULA
BY USING A LIMIT DEFINITION 
OF DERIVATIVE.
IF YOU LOOK 
AT ANY CALCULUS TEXTBOOK
MOST OF THE PROOFS WILL BE 
IN THERE.
IF YOU'RE CURIOUS, I WOULD 
ENCOURAGE YOU TO LOOK AT IT.
OKAY. THE DERIVATIVE OF X 
TO THE POWER OF 5
WOULD BE EQUAL TO 5, 
WILL BE THE COEFFICIENT,
AND THE NEW EXPONENT 
WILL BE 5 - 1.
SO OUR DERIVATIVE WOULD BE 5X 
TO THE POWER OF 4.
NOW THESE NEXT 2 EXAMPLES DO NOT 
FIT THE FORM OF THIS FORMULA,
SO WHAT WE'RE GOING TO HAVE TO 
DO IS REWRITE THIS
SO THAT IT DOES FIT.
WHAT I MEAN BY THAT IS WE HAVE 
TO RECOGNIZE THAT THE CUBIT OF X
IS THE SAME 
AS X TO THE 1/3 POWER.
SO OUR DERIVATIVE WOULD BE 
1/3X TO THE 1/3 - 1
WHICH WOULD BE 
1/3X TO THE POWER OF -2/3.
SIMPLIFYING THIS, 
WE WOULD HAVE 1/3.
NOW IF I MOVE THIS 
TO THE DENOMINATOR
OF COURSE IT WOULD BE BECOME
X TO THE 2/3 POWER.
OUR ORIGINAL FUNCTION 
WAS IN RADICAL FORM,
SO I'M GOING TO CONVERT IT BACK 
INTO RADICAL FORM.
SO THIS WOULD BE 
THE CUBE ROOT OF X SQUARED.
THIS NEXT DERIVATIVE, WE NEED TO 
MOVE THIS UP INTO THE NUMERATOR,
SO WE WOULD HAVE 
X TO THE POWER OF -2.
NOW, I'LL FIND THE DERIVATIVE 
OF THIS.
IT WOULD BE 
-2 x X TO THE POWER OF -2 - 1
WHICH WOULD BE EQUAL TO -2 
x X TO THE POWER OF -3.
AGAIN, SIMPLIFYING,
MOVING THE X TO THE -3
INTO THE DENOMINATOR WOULD 
CHANGE THE SIGN OF THE EXPONENT,
SO OUR FINAL ANSWER WOULD BE -2 
DIVIDED BY X TO THE 3rd.
OKAY, THE NEXT 2 RULES,
THE NEXT RULE IS 
THE CONSTANT MULTIPLE RULE.
IF F IS A DIFFERENTIABLE 
FUNCTION AND C IS A REAL NUMBER
THEN C x F IS 
ALSO DIFFERENTIABLE,
AND THE DERIVATIVE WOULD JUST BE 
C x THE DERIVATIVE OF F OF X.
SO WHAT THIS FORMULA IS SAYING, 
IF I WANT TO FIND THE DERIVATIVE
OF 3X SQUARED WITH RESPECTS 
TO X,
IT WOULD BE 3 
x THE DERIVATIVE OF X SQUARED.
WELL, THE DERIVATIVE 
OF X SQUARED WOULD BE
2X TO THE POWER OF 2 -1, 
OF COURSE WOULD GIVE US 1,
SO OUR DERIVATIVE = 6X.
ON THIS NEXT EXAMPLE
THE FIRST THING I SEE 
IS THERE'S NO EXPONENT ON X.
OF COURSE, IT'S AN IMPLIED 1,
SO I'LL GO AHEAD AND PUT THAT 
IN THERE.
SO THE DERIVATIVE OF THIS WOULD 
BE EQUAL TO
1/2 x THE DERIVATIVE 
OF X TO THE 1st
WHICH WOULD BE 1 x X TO THE--
1 - 1 WOULD BE 0.
SO WE KNOW THAT 1/2 x 1 = 1/2 
x X TO THE 0,
BUT OF COURSE X TO THE 0 
IS ALSO 1.
SO OUR DERIVATIVE = 1/2.
THE NEXT RULE, 
THE SUM AND DIFFERENCE RULES,
THE DERIVATIVE OF F OF X 
+/- G OF X WITH RESPECTS TO X
= THE DERIVATIVE OF F 
+/- THE DERIVATIVE OF G.
SO ESSENTIALLY,
IF I WANT TO FIND THE DERIVATIVE 
OF THIS EXAMPLE,
I JUST NEED TO FIND 
THE DERIVATIVE OF EACH TERM.
SO IF I WANTED TO TAKE THE TIME
I COULD REWRITE THIS 
AS 3 INDIVIDUAL DERIVATIVES.
OF COURSE, 
LOOKING AT THE SECOND TERM,
I'M GOING TO HAVE TO REWRITE 
THIS IN RATIONAL EXPONENT FORM
AND ALSO MOVE IT UP 
TO THE NUMERATOR,
SO THIS IS X TO THE 1/2 
IN THE DENOMINATOR.
IF I MOVE IT UP,
WE'RE GOING TO HAVE 5X 
TO THE POWER OF -1/2
- THE DERIVATIVE OF 8.
NOW SOMETIMES WE'LL SKIP THIS 
STEP, BUT I'LL SHOW IT HERE.
NOW WE'LL APPLY THE POWER RULE 
TO EACH OF THESE TERMS
TO FIND THE DERIVATIVE,
SO HERE WE'D HAVE 2 x 7X 
TO THE POWER OF 6 - 5
x -1/2X TO THE POWER OF -1/2
- 1 WOULD BE -3/2.
THE DERIVATIVE OF 8, OF COURSE, 
WOULD BE 0. LET'S SIMPLIFY.
WE'D HAVE 14X TO THE 6th 
+ 5/2X TO THE -3/2 POWER.
AGAIN, WE CAN MOVE THIS 
TO THE DENOMINATOR
TO MAKE IT A POSITIVE EXPONENT.
SO TO SIMPLIFY THIS, I'LL MOVE 
THE X TO THE DENOMINATOR
TO MAKE IT X 
TO THE POWER OF 3/2.
OKAY. I HOPE THAT HELPS EXPLAIN 
SOME BASIC DERIVATIVE RULES.
THANK YOU FOR WATCHING.
MY NEXT VIDEO WILL DEAL WITH 
APPLICATIONS OF THESE RULES.
