- WE'RE GIVEN F OF (X, Y)
AND ASKED TO FIND 
THE PARTIAL DERIVATIVES.
WE FIRST HAVE 
THE PARTIAL DERIVATIVE OF F
WITH RESPECT TO X,
AND THEN THE PARTIAL DERIVATIVE 
OF F WITH RESPECT TO Y.
TO DETERMINE THE PARTIAL 
DERIVATIVE OF F
WITH RESPECT TO X 
WE'LL CONSIDER Y A CONSTANT
AND DIFFERENTIATE 
WITH RESPECT TO X.
AND THEN TO DETERMINE 
THE PARTIAL DERIVATIVE OF F
WITH RESPECT TO Y 
WE'LL CONSIDER X A CONSTANT
AND DIFFERENTIATE 
WITH RESPECT TO Y.
SO FOR THIS PARTIAL 
WITH RESPECT TO X,
WE'RE TREATING Y AS A CONSTANT
AND DIFFERENTIATING 
WITH RESPECT TO X.
SO THE DERIVATIVE OF -2X 
TO THE 6th WITH RESPECT TO X
WOULD BE -12X TO THE 5th 
+ THE DERIVATIVE OF 6XY SQUARED,
WE'RE TREATING Y SQUARED 
AS A CONSTANT
AND DIFFERENTIATING 
WITH RESPECT TO X.
THE DERIVATIVE OF 6X WOULD BE 6. 
SO WE HAVE + 6Y SQUARED.
AND THEN THE DERIVATIVE 
OF 5Y TO THE 3rd
WITH RESPECT TO X WOULD BE 0,
AGAIN, BECAUSE WE'RE TREATING Y 
AS A CONSTANT.
SO THIS WOULD BE THE PARTIAL 
DERIVATIVE WITH RESPECT TO X.
AND NOW FOR THE PARTIAL 
DERIVATIVE WITH RESPECT TO Y,
WE'RE TREATING X AS A CONSTANT
AND DIFFERENTIATING 
WITH RESPECT TO Y.
AND THEREFORE, 
THE DERIVATIVE OF -2X TO THE 6th
WITH RESPECT TO Y 
WOULD BE 0, AGAIN,
BECAUSE X IS A CONSTANT.
AND THEN PLUS THE DERIVATIVE 
OF 6XY SQUARED
WITH RESPECT TO Y.
AND SINCE THE DERIVATIVE OF Y 
SQUARED IS 2Y
WE'D HAVE 6X x 2Y OR 12XY,
AND THEN PLUS THE DERIVATIVE 
OF 5Y TO THE 3rd
WITH RESPECT TO Y, 
WHICH WOULD BE 15Y SQUARED.
THIS WOULD BE THE PARTIAL 
DERIVATIVE WITH RESPECT TO Y.
BUT BEFORE WE GO, 
LETS QUICKLY REVIEW
WHAT THESE FUNCTIONS WILL TELL 
US ABOUT THE FUNCTION F OF X, Y.
THE FIRST PARTIAL DERIVATIVE 
OF A FUNCTION OF TWO VARIABLES
TELLS US THE SLOPE WITH 
A TANGENT LINE
AT A GIVEN POINT IN EITHER 
THE X OR Y DIRECTION
BASED UPON WHICH 
PARTIAL DERIVATIVE.
THIS WILL MEASURE THE RATE 
OF CHANGE OF Z
OR THE FUNCTION 
WITH RESPECT TO X
OR THE RATE OF CHANGE OF Z OR 
THE FUNCTION WITH RESPECT TO Y,
AGAIN DEPENDING ON WHICH PARTIAL 
DERIVATIVE WE'RE REFERRING TO.
I HOPE YOU FOUND THIS HELPFUL.
