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Partial Differentiation (continued)

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Higher Derivatives of Compound Functions

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

fract{∂~f,∂~s} _ = _ fract{∂~f,∂~u}fract{∂~u,∂~s} _ + _ fract{∂~f,∂~v}fract{∂~v,∂~s}

etc., so

fract{∂^2~f,∂~s^2} _ _ = _ _ fract{∂,∂~s}rndb{fract{∂~f,∂~u}fract{∂~u,∂~s}} _ + _ fract{∂,∂~s}rndb{fract{∂~f,∂~v}fract{∂~v,∂~s}}

_ _ _ _ _ _ = _ _ fract{∂,∂~s}rndb{fract{∂~f,∂~u}}fract{∂~u,∂~s} _ + _ fract{∂~f,∂~u}fract{∂^2~u,∂~s^2} _ + _ fract{∂,∂~s}rndb{fract{∂~f,∂~v}}fract{∂~v,∂~s} _ + _ fract{∂~f,∂~v}fract{∂^2~v,∂~s^2}

Now

fract{∂,∂~s}rndb{fract{∂~f,∂~u}} _ _ = _ _ fract{∂^2~f,∂~u^2}fract{∂~u,∂~s} _ + _ fract{∂^2~f,∂~v∂~u}fract{∂~v,∂~s} , _ _ _ _ etc.

so

fract{∂^2~f,∂~s^2} _ _ = _ _ rndb{fract{∂^2~f,∂~u^2}fract{∂~u,∂~s} _ + _ fract{∂^2~f,∂~v∂~u}fract{∂~v,∂~s}}fract{∂~u,∂~s} _ + _ fract{∂~f,∂~u}fract{∂^2~u,∂~s^2} _ + _ rndb{fract{∂^2~f,∂~v^2}fract{∂~v,∂~s} _ + _ fract{∂^2~f,∂~u∂~v}fract{∂~u,∂~s}}fract{∂~v,∂~s} _ + _ fract{∂~f,∂~v}fract{∂^2~v,∂~s^2}

fract{∂^2~f,∂~s^2} _ _ = _ _ fract{∂^2~f,∂~u^2}script{rndb{fract{∂~u,∂~s}},,,2,} _ + _ fract{∂^2~f,∂~v^2}script{rndb{fract{∂~v,∂~s}},,,2,} _ + _ fract{∂~f,∂~u}fract{∂^2~u,∂~s^2} _ + _ fract{∂~f,∂~v}fract{∂^2~v,∂~s^2} _ + _ 2 fract{∂^2~f,∂~u∂~v}fract{∂~u,∂~s}fract{∂~v,∂~s}

Homogenous Functions

A function _ ~f ( ~x , ~y ) _ of two variables is said to be a #~{homogenous function} of degree ~m, if

~f ( ~K~x , ~K~y ) _ == _ ~K^~m ~f ( ~x , ~y ) , _ _ any ~K

Euler's Theorem

If _ ~f ( ~x , ~y ) _ is a homogenous function of degree ~m, then

~x fract{∂~f,∂~x} _ + _ ~y fract{∂~f,∂~y} _ = _ ~m ~f

#{Proof}

Implicit Functions

Sometimes we have a function of the form _ ~f ( ~x , ~y ) = 0 , _ where ~y is a function of ~x, and we wish to find _ d~y/d~x , _ d&powtwo.~y/d~x&powtwo. , _ etc. The conventional way of doing this is to differentiate ~f ( ~x , ~y ( ~x ) ) with respect to ~x to obtain a function in ~x , ~y and d~y/d~x, from which we can solve to get d~y/d~x. An alternative is to use the chain rule (with ~t = ~x), where _ ~f ( ~x , ~y ( ~x ) ) = 0 :

fract{d~f,d~x} _ = _ fract{∂~f,∂~x}fract{d~x,d~x} _ + _ fract{∂~f,∂~y}fract{d~y,d~x} _ = _ 0

=>

fract{d~y,d~x} _ = _ - fract{∂~f ./ ∂~x,∂~f ./ ∂~y} _ = _ - fract{~{ _ f}_~x,~{ _ f}_~y}

#{Example}

~x^2 + ~y cos ~x + 2 ~x cos ~y _ = _ 0

then

fract{d~y,d~x} _ = _ - fract{2~x - ~y sin ~x + 2 cos ~y,cos ~x - 2 ~x sin ~y}