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- 1. Partial derivative z = f ( x, y ) ϕ ( t ) = f ( a, t ) tan α = α a b dϕ ( t ) dt |t =b
- 2. Let f ( x1 , x2 ,K , xn ) be defined on M ⊆ En and let ( a1 , a2 ,K , an ) ∈ M . Define ϕ i ( t ) = f ( a1 , a2 ,K , ai −1 , t , ai +1 ,K xn ) If ϕ i ( t ) has a derivative at t = ai , we call its value the partial derivative of f ( x1 , x2 ,K , xn ) by xi at ( a1 , a2 ,K , an ) .
- 3. We use several different ways to denote the partial derivative of f ( x1 , x2 ,K , xn ) by xi at ( a1 , a2 ,K , an ) : ∂f ( x1 , x2 ,K , xn ) ∂xi |[ x1 , x2 ,K, xn ] =[ a1 ,a2 ,K,an ] ∂f ( a1 , a2 ,K , an ) ∂xi f x'i ( x1 , x2 ,K , xn ) |[ x1 , x2 ,K, xn ] =[ a1 ,a2 ,K, an ] f x'i ( a1 , a2 ,K , an ) Note that the symbol ∂ reads "d" and is not identical to the Greeek delta δ !!!
- 4. f x'i ( x1 , x2 ,K , xn ) in n variables that assigns to each The function ( a1 , a2 ,K , an ) f ( x1 , x2 at which the partial,K , xn ) derivative of exists the value of such a partial derivative is then called the f ( x1 , x ,K , xn ) partial derivative of 2 by xi Sometimes this function is also denoted ∂f ( x1 , x2 ,K , xn ) ∂xi
- 5. When calculating such a partrial derivative, we use the following practical approach: When calculating the partial derivative of f ( x1 , x2 ,K , xn ) by xi, we think of every variable other than xi as of a constant parameter and treat it as such. Then we actually calculate the "ordinary" derivative of a function in one variable.
- 6. Calculate all the partial derivatives of the following functions f ( x, y, z ) = 3 x 3 yz 4 − 7 xz + 12 y 5 z 2 f ( x, y, z ) = ln x 2 + y 2 + z 2 x arctan y f ( x, y ) = x2 + y 2
- 7. Partial derivatives of higher orders If a partial derivative is viewed as a function it may again be differentiated by the same or by a different variable to become a partial derivative of a higher order. Theoretically, there may be a partial derivative of an arbirary order if it exists. f x''i x j ( x1 , x2 ,K , xn ) , f x''i xi ( x1 , x2 ,K , xn ) , etc. Notation: ∂ 2 f ( x1 , x2 ,K , xn ) ∂ 2 f ( x1 , x2 ,K , xn ) , , etc. 2 ∂xi ∂x j ∂xi
- 8. ∂ f ( x1 , x2 ,K , xn ) 2 ∂xi ∂x j ∂ f ( x1 , x2 ,K , xn ) 2 = ? ∂x j ∂xi
- 9. Schwartz' theorem Let X = [ x0 , y0 ] be an internal point of the domain of f ( x, y ) '' and let, in a neighbourhood N ( X , δ ) of X, f yx ( x, y ) exist '' and be continuus at X = [ x0 , y0 ] . Then f xy ( x, y ) exists and '' '' f xy ( x0 , y0 ) = f yx ( x0 , y0 ) Similar assertions also hold for functions in more than two variables and for higher order derivatives.
- 10. Schwartz' theorem Let X = [ x0 , y0 ] be an internal point of the domain of f ( x, y ) '' and let, in a neighbourhood N ( X , δ ) of X, f yx ( x, y ) exist '' and be continuus at X = [ x0 , y0 ] . Then f xy ( x, y ) exists and '' '' f xy ( x0 , y0 ) = f yx ( x0 , y0 ) Similar assertions also hold for functions in more than two variables and for higher order derivatives.

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