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Partial Derivatives
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  1. Partial Derivatives • Differentiable Function: A function z = f(x,y) is differentiable at (x0, y0) if fx(x0, y0) and fy(x0, y0) exist and Dz satisfies an equation of the form Dz = fx(x0, y0)Dx + fy(x0, y0)Dy + e1Dx + e2Dy, in which each of e1, e2 –> 0 as both Dx, Dy –> 0. We call fdifferentiable if it is differentiable at every point in its domain. • Theorem 1: Properties of Limits of Functions of Two Variables: The following rules hold if L, M, and k are real numbers and lim(x,y) –> (x0,y0)f(x,y) = L and lim(x,y)–> (x0,y0)g(x,y) = M. • Sum Rule: lim(x,y) –> (x0,y0) (f(x,y) + g(x,y)) = L + M • Difference Rule: lim(x,y) –> (x0,y0) (f(x,y) – g(x,y)) = L – M • Product Rule: lim(x,y) –> (x0,y0) (f(x,y) •g(x,y)) = L•M • Constant Multiple Rule: lim(x,y) –> (x0,y0)kf(x,y) = kL (any number k) • Quotient Rule: lim(x,y) –> (x0,y0) (f(x,y) / g(x,y)) = L / M M 0 • Power Rule: If r and s are integers with no common factors, and s  0, then lim(x,y) –> (x0,y0) (f(x,y))r/s = Lr/s provided Lr/s is a real number. (If s is even, we assume L > 0.) • Two-Path Test for Nonexistence of a limit: If a function f(x,y) has different limits along two different paths as (x,y) approaches (x0,y0), then lim(x,y) –> (x0,y0)f(x,y) does not exist. [Note: most often used when (x0,y0) is the point (0,0). The different paths are defined as y = mx. By evaluating lim(x,y) –> (0, 0)f(x,mx), if m remains in the result, then the limit varies along different paths varies as a function of m and therefore does not exist.] • Continuity of Composites: If f is continuous at (x0,y0) and g is a single-variable function continuous at f(x0,y0), then the composite function h = g o f defined by h(x,y) = g(f(x,y)) is continuous at (x0,y0). • Theorum 2: The Mixed Derivative Theorum: If f(x,y) and its partial derivatives fx, fy, fxy, and fyxare defined throughout an open region containing point (a,b) and all are continuoous at (a,b) then fxy(a,b) = fyx(a,b). • Theorum 3: The Incremental Theorum for Functions of Two Variables: Suppose that the first partial derivatives of f(x,y) are defined throughout an open region R containing the point (x0, y0) and that fxand fy are continuous at (x0, y0). Then the change Dz = fx(x0 + Dx, y0 + Dy) – f(x0, y0) in the value of f that results from moving from (x0, y0) to another point (x0 + Dx, y0 + Dy) in R satisfies an equation of the form Dz = fx(x0, y0)Dx + fy(x0, y0)Dy + e1Dx + e2Dy, in which each of e1, e2 –> 0 as bothDx, Dy –> 0 • Corollary of Theorum 3: Continuity of Partial Derivatives Implies Differentiability: If the partial derivatives fx and fy of a function f(x,y) are continuous throughtout an open region R, then f is differentiable at every point in R. • Theorum 4: Differentiability Implies Continuity: If a function f(x,y) is differentiable at (x0, y0), then f is continuous at (x0, y0). Definitions: Function of n Independent Variables: Suppose D is a set of n-tuples of real numbers (x1, x2, x3, …, xn). A real-valued functionf on D is a rule that assignes a unique (single) real number w = f (x1, x2, x3, …, xn) to each element in D. The set D is the function’s domain. The set of w-values taken on by f is the function’s range. The symbol w is the dependent variable of f, and f is said to be a function of the nindependent variablesx1 to xn. We also call the xj’’s the function’s input variables and call w the function’s output variable. Interior and Boundary Points, Open, Closed: A point (x0, y0) in a region (set) R in the xy-plane is an interior point of R if it is the center of a disk of positive radius that lies entirely in R. A point (x0, y0) is a boundary point of R if every disk centered at (x0, y0) contains points that lie outside of R as well as points that lie in R. (The boundary point itself need not belong to R.) The interior points of a region, as a set, make up the interior of the region. The region’s boundary points make up its boundary. A region is open if it consists entirely of interior points. A region is closed if it contains all of its boundary points. Bounded and Unbounded Regions in the Plane: A region in the plane is bounded if it lies inside a disk of fixed radius. A region is unbounded if it is not bounded. Level Curve, Graph, Surface: The set of points in the plane where a function f(x,y) has a constant value f(x,y) = c is called a level curve of f. The set of all points (x, y, f(x,y)) in space, for (x,y) in the domain of f, is called the graph of f. The graph of f is also called the surfacez =f(x,y). Level Surface: The set of points (x,y,z) in space wher a function of three independent variables has a constant value f(x,y,z) = c is called a level surface of f. Interior and Boundary Points for Space Regions: A point (x0, y0, z0) in a region R in space is an interior point of R if it is the center of a solid ball that lies entirely in R. A point (x0, y0, z0) is a boundary point of r if every sphere centered at (x0, y0, z0) encloses points that lie outside of R as well as points that lie inside R. The booundary of R is the set of boundary points of R. A region is open if it consists entirely of interior points. A region is closed if it contains its entire boundary. Limit of a function of Two Variables: We say a function f(x,y) approaches the limitL as (x,y) approaches (x0,y0), and write lim f(x,y) = L if , for every number e > 0, there exists a correspondingnumber d > 0 such that far all (x,y) in the domain of f, |f(x,y) – L| < e whenever 0< (x – x0)2 + (y – y0)2 < d. Continuous Function of Two Variables: A function f(x,y) is continuous at the point (x0,y0) if (1)f is defined at (x0,y0), (2) lim(x,y) –> (x0,y0)f(x,y) exists, (3) lim(x,y) –> (x0,y0)f(x,y) = f(x0,y0). A function is continuous if it is continuous at every point of its domain. Partial Derivative with Respect to x: The partialderivative of f(x,y) with respect to x at the point (x0,y0) is provided the limit exists. Partial Derivative with Respect to y: The partialderivative of f(x,y) with respect to y at the point (x0,y0) is provided the limit exists. [Note: In many cases, df/dx  df/dy ] dff(x0+h,y0) – f(x0,y0) dxh 0 h dff(x0,y0+h) – f(x0,y0) dyh 0 h lim lim = = fxx = = ( ), fyy = = ( ) (x0,y0) (x0,y0) d2f ddf d2f d df dx2dx dx dy2dy dy (x,y) (x0,y0) fxy== ( )= fyx= = ( ) d2f ddf d2f d df dx dy dx dy dy dx dy dx • Notes: • When calculating df/dx, any y’s in the equation are treated as constants when taking the derivative. Similarly, when calculating df/dy, any x’s in the equation are treated as constants when taking the derivative. • Second order partial derivatives • F(x,y,z) = 0 implicitly defines a function z = G(x,y) • Implicit Partial Differentiation, e.g. find dz/dx of xz – y ln z = x + y. Treat y’s like constants. fx = fy = dxz – dy ln z = dx + dy xdz + zdx – ydln z = dx + dy xdz + z – y dz = dx + 0 dz = 1 – z dx dx dx dx dx dx dx dx dx dx zdx dx dx x – y/z

  2. Partial Derivatives (continued) • Theorum 5: Chain rule for Functions of Two Independent Variables: If w = f(x,y) has continuous partial derivatives fx and fy and if x = (t) and y = y(t) are differentiable functions of t, then the composite function w = f(x(t), y(t)) is a differentiable function of t and df/dt = fx(x(t), y(t)) * x’(t) + fy(x(t), y(t)) * y’(t) , or • Theorum 6: Chain Rule for Functions of Three Independent Variables: If w = f(x,y,z) is differentiable and x, y, and z are differentiable functions of t, then w is a differentiable function of t and • Theorum 7: Chain Rule for Two Independent Variables and Three Intermediate Variables: Suppose that w = f(x,y,z), x = g(r,s), y = h(r,s), and z = k(r,s). If all four functions are differentiable, then w has partial derivatives with respect to r and s given by the formulas • Note the following extensions of Theorum 7: If w = f(x,y), x = g(r,s), and y = h(r,s) then • If w = f(x) and x = g(r,s), then • Theorum 8: A Formula for Implicit Differentiation: Suppose that F(x,y) is differentiable and that the equation F(x,y) = 0 defines y as a differentiable function of x. Then at any point where Fy 0, dy/dx = – Fx/Fy • Definition: Directional Derivative: The directional derivative of f at P0(x0, y0) in the direction of the unit vector u = u1i + u2j is the number • provided the limit exists. • Definition: Gradient Vector: The gradient vector (gradiant) of f(x,y) at a point P0(x0, y0) is the vector obtained by evaluating the partial derivatives of f at P0. • Theorum 9: The Directional Derivative Is a Dot Product: If f(x,y) is differentiable in an open • region containing P0(x0, y0), then , the dot product of the gradient f at P0 and u. [Note: Duf = fu = | f | |u| cos q = | f | cos q • Properties of the Directional Derivative Duf = f u = | f | cosq: • The function f increases most rapidly when cos q = 1 or when u is the direction of f. That is, at each point P in its domain, f increases most rapidly in the direction of the gradient vector f at P. The derivative in this direction is Duf = | f | cos (0) = | f |. • Similarly, f decreases most rapidly in the direction of –f . The derivative in this direction is Duf = | f | cos (p) = – | f |. • Any direction u orthogonal to a gradient f 0 is a direction of zero change in f because q then equals p/2 and Duf = | f | cos (p/2) = | f | * 0 = 0 also written as dw dfdx df dy df dz dt dx dt dy dt dz dt = + + dw dfdx df dy dt dx dt dy dt dw dw dx dw dy dt dx dt dy dt = + = + dw dw dx dw dy dw dz ds dx ds dy ds dz ds dw dw dx dw dy dw dz dr dx dr dy dr dz dr = + + = + + dw dw dx dw dy dr dx dr dy dr dw dw dx dw dy ds dx ds dy ds = + = + dw dw dx dr dx dr dw dw dx ds dx ds = = df f(x0 + su1, y0 +su2) – f(x0, y0) ds x  0 s ( )u,P0 (Duf)P0 = = lim df ds ( )u,P0 ∆ (Duf)P0 = = ( f)P0u ∆ ∆ ∆ df dfdx dy ∆ ∆ ∆ f = i + j ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆