- 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.