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Unformatted text preview: Differentiation  a quick review The derivative generalizes to multiple dimensions in a straightforward way. x i denotes all variable other than i . Definition Suppose f : R n R . The partial derivative of a function: x i f ( x ) = f i ( x ) = lim h f ( x i , x i ) f ( x ) h Effectively, you are treating all variables other than i as a constant and taking a normal derivative. ECG 700 Lecture 1  Math Review 1/ 1 Differentiation  a quick review Find the partial derivatives of the following functions? x + y x y 3 x 2 y yxy 1 2 ECG 700 Lecture 1  Math Review 2/ 1 The Differential Let x be a (small) increment and f : R R . Then the increment of y is defined to be: y = f ( x + x ) f ( x ) dx is a variable representing any change in x . The differential of y is defined to be: dy = f ( x ) dx ECG 700 Lecture 1  Math Review 3/ 1 The Differential The way I remember it (although this isnt exactly right) is: f ( x ) = dy dx therefore f ( x ) dx = dy dx * dx = dy Alternatively: f ( x ) = lim x f ( x + x ) f ( x ) x = lim x y x Therefore, for small x , y = y x x f ( x ) dx ECG 700 Lecture 1  Math Review 4/ 1 The Differential The differential is related to the tangent line. In particular, dy y ECG 700 Lecture 1  Math Review 5/ 1 The Differential  Multiple Dimensions We can define the differential analogously for larger dimensions. Suppose f : R n R dx 1 , dx 2 , . . . , dx n are random variables. Define the vector = < dx 1 , dx 2 , . . . , dx n > . f = f ( x + ) f ( x ) The total differential is defined to be: df = n X i x i f ( x ) * dx i ECG 700 Lecture 1  Math Review 6/ 1 The Differential  Multiple Dimensions Definition The gradiant of f : R n R is the vector: f = < x 1 f ( x ) , x 2 f ( x ) , . . . , x n f ( x ) > Similar to before: f f Note that f and are ndimensional vectors, and is the inner product....
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This note was uploaded on 10/22/2009 for the course ECG 700 taught by Professor Morrill during the Fall '09 term at N.C. State.
 Fall '09
 Morrill

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