PARTIAL DERIVATIVES DANG VAN CUONG I HC DUY

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PARTIAL DERIVATIVES DANG VAN CUONG ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY

PARTIAL DERIVATIVES DANG VAN CUONG ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

PARTIAL DERIVATIVES In general, if f is a function of two variables x and

PARTIAL DERIVATIVES In general, if f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. § Then, we are really considering a function of a single variable x: g(x) = f(x, b) ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

PARTIAL DERIVATIVE If g has a derivative at a, we call it the partial

PARTIAL DERIVATIVE If g has a derivative at a, we call it the partial derivative of f with respect to x at (a, b). We denote it by: fx(a, b) Thus, fx(a, b)=g’(a) where g(x) = f(x, b) ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

PARTIAL DERIVATIVE By the definition of a derivative, we have: So, ĐẠI HỌC DUY

PARTIAL DERIVATIVE By the definition of a derivative, we have: So, ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

PARTIAL DERIVATIVE Similarly, the partial derivative of f with respect to y at (a,

PARTIAL DERIVATIVE Similarly, the partial derivative of f with respect to y at (a, b), denoted by fy(a, b), is obtained by: § Keeping x fixed (x = a) § Finding the ordinary derivative at b of the function G(y) = f(a, y) Thus, ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

NOTATIONS There are many alternative notations for partial derivatives. § For instance, instead of

NOTATIONS There are many alternative notations for partial derivatives. § For instance, instead of fx, we can write f 1 or D 1 f (to indicate differentiation with respect to the first variable) or ∂f/∂x. § However, here, ∂f/∂x can’t be interpreted as a ratio of differentials. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

NOTATIONS FOR PARTIAL DERIVATIVES If z = f(x, y), we write: ĐẠI HỌC DUY

NOTATIONS FOR PARTIAL DERIVATIVES If z = f(x, y), we write: ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

RULE TO FIND PARTIAL DERIVATIVES OF z = f(x, y) Thus, we have this

RULE TO FIND PARTIAL DERIVATIVES OF z = f(x, y) Thus, we have this rule. 1. To find fx, regard y as a constant and differentiate f(x, y) with respect to x. 2. To find fy, regard x as a constant and differentiate f(x, y) with respect to y. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION To give a geometric interpretation of partial derivatives, we recall that the

GEOMETRIC INTERPRETATION To give a geometric interpretation of partial derivatives, we recall that the equation z = f(x, y) represents a surface S (the graph of f). § If f(a, b) = c, then the point P(a, b, c) lies on S. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION By fixing y = b, we are restricting our attention to the

GEOMETRIC INTERPRETATION By fixing y = b, we are restricting our attention to the curve C 1 in which the vertical plane y = b intersects S. § That is, C 1 is the trace of S in the plane y = b. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION Likewise, the vertical plane x = a intersects S in a curve

GEOMETRIC INTERPRETATION Likewise, the vertical plane x = a intersects S in a curve C 2. Both the curves C 1 and C 2 pass through P. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION Notice that the curve C 1 is the graph of the function

GEOMETRIC INTERPRETATION Notice that the curve C 1 is the graph of the function g(x) = f(x, b). § So, the slope of its tangent T 1 at P is: g’(a) = fx(a, b) ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION The curve C 2 is the graph of the function G(y) =

GEOMETRIC INTERPRETATION The curve C 2 is the graph of the function G(y) = f(a, y). § So, the slope of its tangent T 2 at P is: G’(b) = fy(a, b) ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION Thus, the partial derivatives fx(a, b) and fy(a, b) can be interpreted

GEOMETRIC INTERPRETATION Thus, the partial derivatives fx(a, b) and fy(a, b) can be interpreted geometrically as: § The slopes of the tangent lines at P(a, b, c) to the traces C 1 and C 2 of S in the planes y = b and x = a. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION Example 2 If f(x, y) = 4 – x 2 – 2

GEOMETRIC INTERPRETATION Example 2 If f(x, y) = 4 – x 2 – 2 y 2 find fx(1, 1) and fy(1, 1) and interpret these numbers as slopes. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION Example 2 We have: fx(x, y) = -2 x fy(x, y) =

GEOMETRIC INTERPRETATION Example 2 We have: fx(x, y) = -2 x fy(x, y) = -4 y fx(1, 1) = -2 fy(1, 1) = -4 ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION Example 2 The graph of f is the paraboloid z = 4

GEOMETRIC INTERPRETATION Example 2 The graph of f is the paraboloid z = 4 – x 2 – 2 y 2 The vertical plane y = 1 intersects it in the parabola z = 2 – x 2, y = 1. § As discussed, we label it C 1. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION The slope of the tangent line to this parabola at the point

GEOMETRIC INTERPRETATION The slope of the tangent line to this parabola at the point (1, 1, 1) is: Example 2 fx(1, 1) = -2 ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION Example 2 Similarly, the curve C 2 in which the plane x

GEOMETRIC INTERPRETATION Example 2 Similarly, the curve C 2 in which the plane x = 1 intersects the paraboloid is the parabola z = 3 – 2 y 2, x = 1. § The slope of the tangent line at (1, 1, 1) is: fy(1, 1) = – 4 ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

GEOMETRIC INTERPRETATION This is a computer-drawn counterpart to the first figure in Example 2.

GEOMETRIC INTERPRETATION This is a computer-drawn counterpart to the first figure in Example 2. § The first part shows the plane y = 1 intersecting the surface to form the curve C 1. § The second part shows C 1 and T 1. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

FUNCTIONS OF MORE THAN TWO VARIABLES Partial derivatives can also be defined for functions

FUNCTIONS OF MORE THAN TWO VARIABLES Partial derivatives can also be defined for functions of three or more variables. § For example, if f is a function of three variables x, y, and z, then its partial derivative with respect to x is defined as: ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

FUNCTIONS OF MORE THAN TWO VARIABLES It is found by: § Regarding y and

FUNCTIONS OF MORE THAN TWO VARIABLES It is found by: § Regarding y and z as constants. § Differentiating f(x, y, z) with respect to x. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

FUNCTIONS OF MORE THAN TWO VARIABLES If w = f(x, y, z), then fx

FUNCTIONS OF MORE THAN TWO VARIABLES If w = f(x, y, z), then fx = ∂w/∂x can be interpreted as the rate of change of w with respect to x when y and z are held fixed. § However, we can’t interpret it geometrically since the graph of f lies in four-dimensional space. ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

FUNCTIONS OF MORE THAN TWO VARIABLES In general, if u is a function of

FUNCTIONS OF MORE THAN TWO VARIABLES In general, if u is a function of n variables, u = f(x 1, x 2, . . . , xn), its partial derivative with respect to the i th variable xi is: ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

FUNCTIONS OF MORE THAN TWO VARIABLES Then, we also write: ĐẠI HỌC DUY T

FUNCTIONS OF MORE THAN TWO VARIABLES Then, we also write: ĐẠI HỌC DUY T N – DUY TAN UNIVERSITY www. duytan. edu. vn www. dtu. edu. vn www. duet. vn

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