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Lecture1b-2ps

Lecture1b-2ps - Differentiation a quick review The...

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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 0 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

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The Differential Let Δ x be a (small) increment and f : R R . Then the increment of y is defined to be: Δ y = f ( x + Δ x ) - f ( x ) dx is a variable representing any change in x . The differential of y is defined to be: dy = f 0 ( x ) dx ECG 700 Lecture 1 - Math Review 3/ 1 The Differential The way I remember it (although this isn’t exactly right) is: f 0 ( x ) = dy dx therefore f 0 ( x ) dx = dy dx * dx = dy Alternatively: f 0 ( x ) = lim Δ x 0 f ( x + Δ x ) - f ( x ) Δ x = lim Δ x 0 Δ y Δ x Therefore, for small Δ x , Δ y = Δ y Δ x Δ x f 0 ( x ) dx ECG 700 Lecture 1 - Math Review 4/ 1
The Differential The differential is related to the tangent line. In particular, dy Δ 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 Δ = < dx 1 , dx 2 , . . . , dx n > . Δ f = f ( x + Δ) - 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

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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: Δ f ≈ ∇ f · Δ Note that f and Δ are n-dimensional vectors, and · is the inner product. ECG 700 Lecture 1 - Math Review 7/ 1 Implicit Function Theorem Most functions we look at are of the form: y = F ( x 1 , . . . , x n ) We say y is an explicit function of x 1 , . . . , x n .
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Lecture1b-2ps - Differentiation a quick review The...

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