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Partial Differentiation

 
 

Partial Derivatives

If _ ~f ( ~x , ~y ) _ is a function of two variables, then it has a value for every point ( ~x , ~y ) of the Cartesian plane. For any given ~y we have a function of ~x, _ ~f _{(~y)} ( ~x ) , _ which can be differentiated with respect to ~x. We call this derivative the #~{partial derivative} of ~f with respect to ~x :

fract{&partial.~f,&partial.~x} _ #:= _ fract{d ~f _{(~y)},d~x}

If it is necessary to make clear that it is the variable ~y which is being kept constant we will write this as _ {&partial.~f} ./ {&partial.~x} )_~y .

In the same way we define partial derivative of ~f with respect to ~y :

fract{&partial.~f,&partial.~y} _ #:= _ fract{d ~f _{(~x)},d~y}

If ~f is given as a formula in ~x and ~y, then, for example, {&partial.~f} ./ {&partial.~x} can be calculated by differentiating the formula treating ~x as a variable and ~y as a constant.

Note: _ {&partial.~f} ./ {&partial.~x} _ and _ {&partial.~f} ./ {&partial.~y} _ are often written as _ ~f_~x _ and _ ~f_~y _ respectively. Do not confuse these with the notation ~f _{(~y)} which we used above.

#{Example}

~f ( ~x , ~y ) _ = _ 6 ~x^3 ~y^2

~f_~x _ = _ {&partial.~f} ./ {&partial.~x} _ = _ 18 ~x^2 ~y^2

~f_~y _ = _ {&partial.~f} ./ {&partial.~y} _ = _ 12 ~x^3 ~y

Function of Several Variables

In the same way we can calculate partial derivatives of a function of several variables, for example

~f ( ~x , ~y , ~z ) _ = _ ~x^2 ~y ~z + ~y ~z^2

{&partial.~f} ./ {&partial.~x} _ = _ 2 ~x ~y ~z , _ _ {&partial.~f} ./ {&partial.~y} _ = _ ~x^2 ~z + ~z^2 , _ _ {&partial.~f} ./ {&partial.~z} _ = _ ~x^2 ~y + 2 ~y ~z

I.e. in the case of _ {&partial.~f} ./ {&partial.~x} _ both ~y and ~z are considered as constants, etc.

 

Higher Partial Derivatives

If ~f is a function of ~x and ~y, then {&partial.~f} ./ {&partial.~x} and {&partial.~f} ./ {&partial.~y} are also functions of ~x and ~y, so we can define

_ ~f_{~x~x} _ #:= _ fract{&partial.^2~f,&partial.~x^2} _ #:= _ fract{&partial.,&partial.~x}rndb{fract{&partial.~f,&partial.~x}} , _ _ _ _ _ _ ~f_{~y~y} _ #:= _ fract{&partial.^2~f,&partial.~y^2} _ #:= _ fract{&partial.,&partial.~y}rndb{fract{&partial.~f,&partial.~y}}

_ ~f_{~y~x} _ #:= _ fract{&partial.^2~f,&partial.~y&partial.~x} _ #:= _ fract{&partial.,&partial.~y}rndb{fract{&partial.~f,&partial.~x}} , _ _ _ _ _ _ ~f_{~x~y} _ #:= _ fract{&partial.^2~f,&partial.~x&partial.~y} _ #:= _ fract{&partial.,&partial.~x}rndb{fract{&partial.~f,&partial.~y}}

#{Example}

Using the same example as above

~f ( ~x , ~y ) _ = _ 6 ~x^3 ~y^2 _ => _ {&partial.~f} ./ {&partial.~x} _ = _ 18 ~x^2 ~y^2 _ and _ {&partial.~f} ./ {&partial.~y} _ = _ 12 ~x^3 ~y

then

fract{&partial.^2~f,&partial.~x^2} _ = _ 36 ~x ~y^2 , _ _ _ fract{&partial.^2~f,&partial.~y^2} _ = _ 12 ~x^3 ,

fract{&partial.^2~f,&partial.~y&partial.~x} _ = _ 36 ~x^2 ~y , _ _ _ fract{&partial.^2~f,&partial.~x&partial.~y} _ = _ 36 ~x^2 ~y ,

Note that in this case

fract{&partial.^2~f,&partial.~y&partial.~x} _ == _ fract{&partial.^2~f,&partial.~x&partial.~y}

In fact this can always be assumed to be the case.

Chain Rule for Partial Derivatives

If ~f is a function of ~x and ~y, which are in turn functions of a variable ~t , _ ~f _ = _ ~f ( ~x , ~y ) , _ ~x _ = _ ~x ( ~t ) , _ ~y _ = _ ~y ( ~t ) , _ then ~f is also a function of ~t . [ Strictly speaking, we can define the composite function: _ ~f ( ~t ) #:= ~f ( ~x ( ~t ) , ~y ( ~t ) ) .] _ We have:

fract{d~f,d~t} _ = _ fract{&partial.~f,&partial.~x}fract{d~x,d~t} _ + _ fract{&partial.~f,&partial.~y}fract{d~y,d~t}

Similarly, if ~f is a function of ~u and ~v, which in turn are functions of two variables, ~s and ~t, then

fract{&partial.~f,&partial.~s} _ = _ fract{&partial.~f,&partial.~u}fract{&partial.~u,&partial.~s} _ + _ fract{&partial.~f,&partial.~v}fract{&partial.~v,&partial.~s}

fract{&partial.~f,&partial.~t} _ = _ fract{&partial.~f,&partial.~u}fract{&partial.~u,&partial.~t} _ + _ fract{&partial.~f,&partial.~v}fract{&partial.~v,&partial.~t}

Polar Coordinates

Any point ( ~x , ~y ) in the Cartesian plane can be represented by the polar coordinates ( ~r , &theta. ), where

~r = +&sqrt.${ ~x&powtwo. + ~y&powtwo. } , _ &theta. = tan^{-1} ( ~y ./ ~x ), _ -&pi. < &theta. =< +&pi. .

Conversely

~x = ~r cos &theta. , _ ~y = ~r sin &theta.

So if we have ~f ( ~x , ~y ), then

(A) _ _ fract{&partial.~f,&partial.~r} _ = _ cos &theta. fract{&partial.~f,&partial.~x} _ + _ sin &theta. fract{&partial.~f,&partial.~y}

(B) _ _ fract{&partial.~f,&partial.&theta.} _ = _ - ~r sin &theta. fract{&partial.~f,&partial.~x} _ + _ ~r cos &theta. fract{&partial.~f,&partial.~y}

If we have ~f ( ~r , &theta. ), then

fract{&partial.~f,&partial.~x} _ _ = _ _ fract{~x,&sqrt.${ ~x&powtwo. + ~y&powtwo. }} fract{&partial.~f,&partial.~r} _ - _ fract{~y, ~x&powtwo. + ~y&powtwo. } fract{&partial.~f,&partial.&theta.} _ _ = _ _ cos &theta. fract{&partial.~f,&partial.~r} _ - _ fract{sin &theta.,~r} fract{&partial.~f,&partial.&theta.}

fract{&partial.~f,&partial.~y} _ _ = _ _ fract{~y,&sqrt.${ ~x&powtwo. + ~y&powtwo. }} fract{&partial.~f,&partial.~r} _ + _ fract{~x, ~x&powtwo. + ~y&powtwo. } fract{&partial.~f,&partial.&theta.} _ _ = _ _ sin &theta. fract{&partial.~f,&partial.~r} _ + _ fract{cos &theta.,~r} fract{&partial.~f,&partial.&theta.}

Note that the expressions for _ &partial.~f ./ &partial.~x _ and _ &partial.~f ./ &partial.~y _ can also be found by solving for these in (A) and (B) above.