Sec. 14.3: Partial Derivatives. All of the following are ways of representing the derivative. y dx

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1 Math 2204 Multivariable Calc Chapter 14: Partial Derivatives I. Review from math 1225 A. First Derivative Sec. 14.3: Partial Derivatives 1. Def n : The derivative of the function f with respect to the variable x is the function f f ( x+h)-f ( x) whose value at x is f (x) = lim provided the limit exists. h 0 h 2. Notation All of the following are ways of representing the derivative dy dx, y, f df (x), dx, d ( dx f ( x) ), D f, y x 3. Geometric Interpretation The derivative at a point is the slope of the tangent line at that point. 4. Physically, the derivative at a point is the velocity at that point. B. Second Derivative 1. Def n : If f is a differentiable function, the second derivative is the derivative of the first d dy derivative, i.e. dx dx. The second derivative is the rate of change of the first derivative wrt x or you can interpret the second derivative as the rate of change of the rate of change. 2. Notation d 2 y dx, y, f (x), d 2 f 2 dx, d ( f ( x) ), D 2 2 x f, D 2 x y dx

2 II. Partial Derivatives A. A partial derivative is a rate of change or slope of a cross-sectional model (vertical slice of the graph) of a multivariable function. That is, it is the slope of the tangent plane to a point on the surface of the given function. B. Definitions of Partial Derivatives 1. Def n : The partial derivative of f (x, y) with respect to x is = f x (x, y) = lim h 0 f (x + h, y) f (x, y) h, provided the limit exists. It is found by holding y constant and taking the derivative of the function with respect to x. 2. Def n : The partial derivative of f (x, y) with respect to y is = f y (x, y) = lim h 0 f (x, y + h) f (x, y), provided the limit exists. h It is found by holding x constant and taking the derivative of the function with respect to y. NOTE: The definitions of the partial derivatives of functions of more than two independent variables are similar to the definitions for the functions of two variables. C. Notation If z = f (x, y), f x (x, y) = f x = f = f (x, y) = z f y (x, y) = f y = f = z f (x, y) = D. Geometric Interpretation The partial derivatives f x ) and f y ) can be interpreted geometrically as the slopes of the tangent line at P,z 0 ) to the traces C 1 and C 2 of surface S in the planes y = b and x = a.

3 E. Examples of First Partial Derivatives Find and. 1. f (x, y) = 2x 3 + 7xy 2 5y 4 y 2. f (x, y) = tan 1 x 3. f (x, y) = x x 2 + y 2

4 Find, and z. 4. f (x, y,z) = yzln(xy) 5. Use implicit differentiation to find δ z δ x. x 2 y 2 + z 2 2z = 4

5 III. Second-Order Partial Derivatives A. The second partial derivatives give the rate of change or slope of the first partials, i.e., the slope or ROC of the tangent planes of the cross-sectional models in a particular direction. B. When a function f (x, y) is differentiated twice, its second-order partial derivatives are produced f 2 = f = ( f x ) x = f xx in the x direction. It is found by holding y constant and taking the derivative of the first partial with respect to x with respect to x again f 2 = f = ( f ) = f y y yy in the y direction. It is found by holding x constant and taking the derivative of the first partial with respect to y with respect to y again f = f = ( f ) = f y x yx in the x direction. It is found by holding y constant and taking the derivative of the first partial with respect to y with respect to x f = f = ( f x ) y = f xy in the y direction. It is found by holding x constant and taking the derivative of the first partial with respect to x with respect to y f = f yx and 2 f = f xy continuous. are called mixed partials and are equal when f (x, y) is C. Theorem 1: Equality of Mixed Partials If f (x, y) and its partial derivatives f x, f y, f xy, and f yx are defined throughout an open region containing a point ) and are all continuous at ), then f xy ) = f yx ).

6 D. Examples of Second-Order Partial Derivatives 1. Given f (x, y) = ye x2 + x cos(y), find 2 f 2, 2 f 2, 2 f and 2 f. 2. Given g(x, y) = xy + ey y 2 +1, find g yx. Note: By Th m 1, we can switch the order of differentiation.

7 IV. Partial Derivatives of Still Higher Order A. We can define derivatives of still higher order. 3 f 2 = f yyx 4 f 2 2 = f yyxx B. A similar equality of mixed partial result holds for higher order derivatives. We can typically compute derivatives in any order. C. Example Given f (x, y) = xe 1+y2, show that f xyyx = 0. V. Differentiability A. The Increment Theorem 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 ) and that f x and f y are continuous at ). Then the change Δz = f + Δx + Δy) f ) in the value of f that results from moving ) to another point + Δx + Δy) in R satisfies an equation of the form Δz = f x )Δx + f y )Δy + ε 1 Δx + ε 2 Δy, in which each of ε 1, ε 2 0 as both Δx, Δy 0. B. Definition A function z = f (x, y) is differentiable at the point ) if f x ) and f y ) exist and Δz satisfies an equation of the form Δz = f x )Δx + f y )Δy + ε 1 Δx + ε 2 Δy, in which each of ε 1, ε 2 0 as both Δx, Δy 0. We call f differentiable if it is differentiable at every point in its domain. C. Corollary of Theorem If the partial derivatives f x and f y of a function f (x, y) are continuous throughout an open region R, then f (x, y) is differentiable at every point of R. D. Differentiability Implies Continuity If a function f (x, y) is differentiable at ), then f (x, y) is continuous at ).