(1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives.

Transcription

1 Section 11.3 Partial Derivatives (1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives. MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 1 / 14

2 Derivatives and Rates of Change Recall that, for functions of one variable, y = f (x), the derivative f (a) at the point (a, f (a)) is the rate of change of values f (x) at x = a. f (a) is a measure of how fast the y-value changes near x = a. For functions of two variables z = f (x, y) and near a point (a, b), there are infinitely many directions/ways that a point (x, y) can vary near (a, b). How do we characterize the rate of change of f (x, y) at a point (a, b) in the x-direction? MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 2 / 14

3 It turns out the rate of change in two directions encodes the information for the rate of change in any direction. Let (a, b) be a point in the domain D of f (x, y). If we fix y = b and vary x, we get a function g(x) = f (x, b) one variable that lies on the surface of f (x, y). The derivative of g (a) is the slope of the tangent line to the graph of g(x). The graph of g(x) is parameterized by (x, b, g(x)), so the vector 1, 0, g (a) is a tangent vector to the graph. The derivative g (a) is the partial derivative of f (x, y) with respect to the x-variable at (a, b). MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 3 / 14

4 The partial derivative of f (x, y) with respect to x at (a, b) is denoted f x (a, b) = f f (a + h, b) f (a, b) x(a, b) = lim h 0 h The partial derivative of f (x, y) with respect to y at (a, b) is denoted f y (a, b) = f f (a, b + h) f (a, b) y (a, b) = lim h 0 h MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 4 / 14

5 To compute the partial derivative with respect to x, f x, treat the y-variable as a constant and apply the ordinary rules for differentiation. Partial differentiation is not implicit differentiation. Compute the partial derivative with respect to y, f y, in an analogous way. Example: z = xy + ye xy f x (x, y) = z x (x, y) = y + y 2 e xy f y (x, y) = z (x, y) = x + (1 + xy)exy y MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 5 / 14

6 The planes x = a and y = b intersect the surface z = f (x, y) as curves z = f (a, y) and z = f (x, b) (respectively). The partial derivatives are the slopes of the tangent lines to the two curves. f (x, b) is the intersection of f and y = b. The tangent line to x = a has point (a, b, f (a, b)) and directional vector 1, 0, f x (a, b). f (a, y) is the intersection of f and x = a. The tangent line to y = b has point (a, b, f (a, b)) and directional vector 0, 1, f y (a, b). Example: f (x, y) = x(y + 1) x 2 at ( 2, 2). f x (x, y) = y + 1 2x f y (x, y) = x The tangent line to the intersection curve of f (x, y) and the plane y = 2 at ( 2, 2, f ( 2, 2)) is r x (t) = 2 + t, 2, t The tangent line to the intersection curve of f (x, y) and the plane x = 2 at ( 2, 2, 10) is r y (t) = 2, 2 + t, 10 2t MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 6 / 14

7 Implicit Partial Derivatives Find z x and z y implicitly from xy + xz + y 2 z e yz = 1. Solution: Take the partial derivative x through the equation: x (xy) + x (xz) + x (y 2 z) x (eyz ) = x (1) Note that z = z(x, y) is a function of (x, y) implicitly. Thus ( y + z + x z ) + y 2 z x x eyz y z x = 0 In a similar way, z x = (y + z) x + y 2 ye yz z y = zeyz x 2yz x + y 2 ye yz MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 7 / 14

8 The extension of partial derivatives of more than two variables is straightforward. In a three variable function t = f (x, y, z), the partial derivative f z is calculated by differentiating with respect to z, treating x and y as constants. Example: Find f x, f f, and y z for f (x, y, z) = z ln(x 2 + y 2 ) + sin(xz). f x (x, y, z) = f x(x, y, z) = 2xz x 2 + y 2 + z cos(xz) f y (x, y, z) = f y (x, y, z) = 2yz x 2 + y 2 f z (x, y, z) = f z(x, y, z) = ln(x 2 + y 2 ) + x cos(xz) MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 8 / 14

9 Higher Order Partial Derivatives For z = f (x, y), the partial derivatives f x (x, y) and f y (x, y) are again functions of (x, y). We can define partial derivatives partial derivatives of these functions; second order partial derivatives of f (x, y). ( ) f (x, y) = 2 x x x 2 f (x, y) = f xx(x, y) In general, ( ) f (x, y) = 2 x y x y f (x, y) = f yx(x, y) ( ) f (x, y) = 2 y x y x f (x, y) = f xy(x, y) y ( f (x, y) y ) = 2 y 2 f (x, y) = f yy(x, y) n f x 1 x 2... x n = f xn...x 2 x 1 MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 9 / 14

10 Compute f xx, f xy, f yx, f yy for f (x, y) = xe xy + y 2 y(x 2 + 1) Solution: f x (x, y) = e xy + xye xy 2xy f y (x, y) = x 2 e xy + 2y (x 2 + 1) f xx (x, y) = 2ye xy + xy 2 e xy 2y f xy (x, y) = 2xe xy + x 2 ye xy 2x f xy (x, y) = 2xe xy + x 2 ye xy 2x f yy (x, y) = x 3 e xy + 2 Note: Here f xy = f yx. If f xy (x, y) and f yx (x, y) are continuous, then f xy = f yx. MATH 127 (Section 11.3) Partial Derivatives The University of Kansas 10 / 14