- TO DETERMINE THE DERIVATIVE 
OF EACH FUNCTION,
WE'LL BE APPLYING THE POWER RULE 
OF DIFFERENTIATION STATED HERE.
THIS TELLS US THE DERIVATIVE
OF THE X TO THE POWER OF N 
= N x X TO THE POWER OF N - 1,
SO WE MULTIPLY BY 
THE CURRENT EXPONENT,
AND THEN WE SUBTRACT 1 
TO GET THE NEW EXPONENT.
NOTICE FOR THIS FIRST FUNCTION 
N = 7,
SO THE DERIVATIVE OF F = 7 x X 
TO THE 7 - 1 POWER
WHICH SIMPLIFIES TO 7X 
TO THE 6th.
FOR THE SECOND EXAMPLE,
NOTICE HOW WE HAVE A COEFFICIENT  
OR A CONSTANT x A FUNCTION OF X,
AND THIS DERIVATIVE RULE HERE  
IS JUST TELLING US
THAT THE DERIVATIVE OF 
A CONSTANT x THE FUNCTION OF X
= THE CONSTANT x THE DERIVATIVE 
OF THE FUNCTION OF X.
SO FOR F OF X = -8X TO THE 3rd,
F PRIME OF X = -8 
x THE DERIVATIVE OF X TO THE 3rd
WHICH WILL BE 3 
x X TO THE 3 - 1 POWER.
LET'S GO AHEAD 
AND SIMPLIFY THIS.
-24X TO THE 2nd.
SO VERY QUICKLY, YOU'LL PROBABLY 
STOP SHOWING THIS 3 - 1
AND JUST WRITE 2 WHICH I WILL 
DO IN THE NEXT TWO EXAMPLES.
NOW NOTICE FOR THESE 
TWO EXAMPLES
WE HAVE A SUM OR DIFFERENCE 
OF TERMS,
AND THE ONLY DIFFERENCE HERE
IS WE'LL APPLY THE POWER RULE 
TO EACH TERM.
SO IF F OF X = X TO THE 4th 
- 5X SQUARED,
F PRIME OF X 
IS GOING TO BE EQUAL
TO THE DERIVATIVE 
OF X TO THE 4th - 5
x THE DERIVATIVE OF X 
TO THE 2nd.
SO THE DERIVATIVE 
OF X TO THE 4th
WOULD BE 4 x TO THE 4 - 1, 
THAT'LL BE 3rd POWER - 5
x THE DERIVATIVE 
OF X TO THE 2nd.
THAT WOULD BE 2 x X TO THE 2 
- 1, SO WE HAVE A POWER OF 1.
AND NOW WE'LL SIMPLIFY, 
SO WE HAVE 4X TO THE 3rd.
THIS'LL JUST BE - 10X. 
X TO THE 1st = X.
AND NOW FOR THE LAST EXAMPLE 
WE HAVE 3 TERMS,
BUT THE STRATEGY WILL BE 
THE SAME.
F PRIME OF X IS GOING 
TO BE EQUAL
TO THE SUM OF THE DERIVATIVES 
OF EACH TERM,
SO WE'LL HAVE 1/8 
x THE DERIVATIVE OF X TO THE 4th
THAT WILL BE 4 x X TO THE 3rd
+ FOR 2X YOU CAN THINK 
OF THIS AS X TO THE 1st,
SO WE WILL HAVE 2 x 
THE DERIVATIVE OF X TO THE 1st.
IT'LL BE 1X TO THE 1 - 1, 
THAT'S 0 + THE DERIVATIVE OF 5.
THE DERIVATIVE OF ANY CONSTANT 
IS EQUAL TO 0,
SO THE DERIVATIVE OF 5 = 0, 
AND NOW WE'LL SIMPLIFY THIS.
THIS WILL BE 4/8 
OR 1/2X TO THE 3rd,
AND X TO THE 0 = 1, SO WE 
HAVE 2/1 x 1, SO WE HAVE +2.
OKAY. HOPE YOU FOUND 
THESE EXAMPLES HELPFUL.
WE'LL TAKE A LOOK 
AT NEGATIVE EXPONENTS
AND FRACTIONAL EXPONENTS 
IN THE NEXT PART.
