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Unformatted text preview: Ch8/MATH1804/YMC/200910 1 Chapter 8. Partial Differentiation Many functions depend on more than one variable. It is then natural to extend the basic ideas of the calculus of functions of a single variable to functions of several variables. Let us first study functions of two variables, that is, f : R 2→ R . 8.1. Partial Derivatives Definition 8.1 Let f : R 2→ R be a function of two variables, and ( a,b ) ∈ R 2 . We define the partial derivative in the x direction at ( a,b ) as ∂f ∂x ( a,b ) = lim h → f ( a + h, b ) f ( a,b ) h . Similarly define the partial derivative in the y direction at ( a,b ) as ∂f ∂y ( a,b ) = lim k → f ( a,b + k ) f ( a,b ) k . We also use f x and f y to denote ∂f ∂x and ∂f ∂y respectively. Ch8/MATH1804/YMC/200910 2 In other words, the partial derivative ∂f ∂x ( a,b ) is the rate of change of f , at the point ( a,b ) , when we vary only the variable x about a and keep the variable y constant. Similarly, the partial derivative ∂f ∂y ( a,b ) is the rate of change of f , at the point ( a,b ) , when we vary only the variable y about b and keep the variable x constant. Note that it is important to distinguish between the symbol ∂ for a partial derivative and the letter d for an ordinary derivative. Example 8.2 (i) Consider the function f : R 2→ R defined by f ( x, y ) = x 2 + 3 xy . Then ∂f ∂x can be computed by differentiating with respect to x while treating y as a constant. Hence we have ∂f ∂x ( x,y ) = 2 x + 3 y. Similarly, ∂f ∂y is given by ∂f ∂y ( x, y ) = 3 x. Therefore, the values of ∂f ∂x and ∂f ∂y at the point, say (1 , 2) , are given by ∂f ∂x (1 , 2) = 2(1) + 3(2) = 8 and ∂f ∂y (1 , 2) = 3(1) = 3 . (ii) Let f ( x,y ) = ln( x 2 y ) . Then f x ( x, y ) = 1 x 2 y · 2 xy = 2 x and f y ( x,y ) = 1 x 2 y · x 2 = 1 y . (iii) Let g ( u, v ) = 5 e v sin v . Then g u ( u, v ) = 0 and g v ( u, v ) = 5 e v (sin v + cos v ) . Note that g u = 0 since the definition of g is independent of u . (iv) (An Economics Example) A CobbDouglas production function is defined by P ( x, y ) = Ax α y β where P is the total production, x is the labor input and y is the capital input, with A > , < α, β < 1 are constants. Then ∂P ∂x ( x,y ) = αAx α 1 y β and ∂P ∂y ( x, y ) = βAx α y β 1 . Ch8/MATH1804/YMC/200910 3 Definition 8.3 The gradient of a function f ( x, y ) at a point ( a,b ) , denoted by ∇ f ( a,b ) , is the 2 × 1 matrix (vector) ∇ f ( a,b ) = ∂f ∂x ( a,b ) ∂f ∂y ( a,b ) ! . Another common notation for gradient is grad f . Example 8.4 Let f : R 2→ R be a function defined by f ( x, y ) = x 2 + y 2 . Then ∇ f ( x,y ) = ∂f ∂x ( x,y ) ∂f ∂y ( x,y ) !...
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