Partial Derivatives Formulas. KristaKingMath.com
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1 Partial Derivatives Formulas KristaKingMath.com
2 Domain and range of a multivariable function A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D a unique real number denoted by f(x, y). Domain of f: The set D Range of f: The set of values that f takes on Level curves of a multivariable function The level curves of a function f of two variables are the curves with equations f(x, y) = k, where k is a constant (in the range of f). Precise definition of the limit of a multivariable function Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b). Then the limit of f(x, y) as (x, y) approaches (a, b) is L, lim f(x, y) = L (x,y) (a,b) if for all ϵ > 0 there is a corresponding δ > 0 such that if (x, y) D and 0 < (x a) 2 + (y b) 2 < δ then f(x, y) L < ϵ Existence of the limit of a multivariable function If f(x, y) L 1 as (x, y) (a, b) along a path C 1 and f(x, y) L 2 as (x, y) (a, b) along a path C 2 where L 1 L 2, then lim f(x, y) does not exist. (x,y) (a,b) 1
3 Continuity of a multivariable function A function f of two variables is continuous at (a, b) if lim f(x, y) = f(a, b) (x,y) (a,b) In other words, f is continuous at (a, b) if it s limit at (a, b) is equal to the actual value of the function at (a, b) We say f is continuous on D if f is continuous on every point (a, b) in D. Definition of the derivative of a multivariable function Partial derivative with respect to x f x (a, b) = lim h 0 f(a + h, b) f(a, b) h Partial derivative with respect to y f y (a, b) = lim h 0 f(a, b + h) f(a, b) h Notation of partial derivatives If z = f(x, y), then we can write the Partial derivative with respect to x as f x (x, y) = f x = f = f(x, y) = = f 1 = D 1 f = D x f Partial derivative with respect to y as f y (x, y) = f y = f = f(x, y) = = f 2 = D 2 f = D y f Rule for finding partial derivatives of z = f(x, y) To find f x, treat y as a constant and differentiate f(x, y) with respect to x. To find f y, treat x as a constant and differentiate f(x, y) with respect to y. 2
4 Clairaut s theorem for the mixed second-order partial derivative Suppose f is defined on a disk D that contains the point (a, b). If the functions f xy and f yx (the mixed second-order partial derivatives) are both continuous on D, then f xy (a, b) = f yx (a, b) Equation of the tangent plane Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z = f(x, y) at the point P(x 0, y 0, z 0 ) is z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) Differentiability of a multivariable function If z = f(x, y), then f is differentiable at (a, b) if Δz can be expressed as Δz = f x (a, b)δx + f y (a, b)δy + ϵ 1 Δx + ϵ 2 Δy where ϵ 1 and ϵ 2 0 as (Δx, Δy) (0,0). OR If the partial derivatives f x and f y exist near (a, b) and are continuous at (a, b), then f is differentiable at (a, b). Total differential dz = f x (x, y)dx + f y (x, y)dy = dx + dy 3
5 Chain rule for multivariable functions Case 1 - f [g(t), h(t)] Suppose that z = f(x, y) is a differentiable function of x and y, where x = g(t) and y = h(t) are both differentiable functions of t. Then z is a differentiable function of t and dz dt = f dx dt + f dy dt dz dt = dx dt + dy dt Case 2 - f [g(s, t), h(s, t)] Suppose that z = f(x, y) is a differentiable function of x and y, where x = g(s, t) and y = h(s, t) are both differentiable functions of s and t. Then s = s + s t = t + t General version Suppose that u is a differentiable function of n variables x 1, x 2,..., x n and each x i is a differentiable function of m variables t 1, t 2,..., t m. Then u is a function of t 1, t 2,..., t m and u = u 1 t i 1 t i + u 2 2 t i + + u n n t i for each i = 1, 2,, m. Implicit differentiation of a multivariable function F dx dy = = F x F F y 4
6 Partial derivatives for implicit differentiation F = F = F F Directional derivative of a function in two variables The directional derivative of f at (x 0, y 0 ) in the direction of a unit vector u = a, b is D u f(x 0, y 0 ) = lim h 0 f(x 0 + ha, y 0 + hb) f(x 0, y 0 ) h if this limit exists. OR If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector u = a, b and D u f(x, y) = f x (x, y)a + f y (x, y)b OR If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector u = a, b and D u f(x, y) = f(x, y) u where f(x, y) is the gradient of the function and u is the unit vector. 5
7 Directional derivative of a function in three variables The directional derivative of f at (x 0, y 0, z 0 ) in the direction of a unit vector u = a, b, c is D u f(x 0, y 0, z 0 ) = lim h 0 f(x 0 + ha, y 0 + hb, z 0 + hc) f(x 0, y 0, z 0 ) h if this limit exists. OR If f is a differentiable function of x, y and z, then f has a directional derivative in the direction of any unit vector u = a, b, c and D u f(x, y, z) = f x (x, y, z)a + f y (x, y, z)b + f z (x, y, z)c OR If f is a differentiable function of x, y and z, then f has a directional derivative in the direction of any unit vector u = a, b, c and D u f(x, y, z) = f(x, y, z) u where f(x, y, z) is the gradient of the function and u is the unit vector. Gradient of a multivariable function If f is a function of two variables x and y, then the gradient of f is the vector function f defined by f(x, y) = f x (x, y), f y (x, y) = f i + f j Gradient vector of a multivariable function f = f x, f y, f z = f i + f j + f k 6
8 Maximizing the directional derivative Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative D u f(x) is f(x) and it occurs when u has the same direction as the gradient vector f(x). Tangent plane to the level surface F x (x 0, y o, z 0 )(x x 0 ) + F y (x 0, y o, z 0 )(y y 0 ) + F z (x 0, y o, z 0 )(z z 0 ) = 0 Local and global extrema of a multivariable function For a function of two variables x and y, If f(x, y) f(a, b) when (x, y) is near (a, b), then f has a local maximum at (a, b) and f(a, b) is a local maximum value, unless the inequality is true for all points (x, y) in the domain of f, in which case f has an absolute maximum at (a, b). If f(x, y) f(a, b) when (x, y) is near (a, b), then f has a local minimum at (a, b) and f(a, b) is a local minimum value, unless the inequality is true for all points (x, y) in the domain of f, in which case f has an absolute minimum at (a, b). 7
9 Second derivatives test Suppose the second partial derivatives of f are continuous on a disk with center (a, b) and suppose that f x (a, b) = 0 and f y (a, b) = 0 [that is, (a, b) is a critical point of f]. Let D = D(a, b) = f xx (a, b)f yy (a, b) [ f xy (a, b)] 2 If D > 0 and f xx (a, b) > 0, then f(a, b) is a local minimum If D > 0 and f xx (a, b) < 0, then f(a, b) is a local maximum If D < 0 and f(a, b) is not a local maximum or minimum ((a, b), f(a, b)) is called a saddle point If D = 0, the test is inconclusive It can t be used to characterize the critical point ((a, b), f(a, b)) Extreme value theorem for multivariable functions If f is continuous on a close, bounded set D in IR 2, then f attains an absolute maximum value f(x 1, y 1 ) and an absolute minimum value f(x 2, y 2 ) at some points (x 1, y 1 ) and (x 2, y 2 ) in D. Steps to identify global extrema To find the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: Find the values of f at the critical points of f in D. Find the extreme values of f on the boundary of D. The largest of the values is the absolute maximum value; the smallest of these values is the absolute minimum value. 8
10 Method of lagrange multipliers To find the maximum and minimum values of f(x, y, z) subject to the constraint g(x, y, z) = k [assuming that these extreme values exist and g 0 on the surface g(x, y, z) = k]: Find all values of x, y, z and λ such that f(x, y, z) = λ g(x, y, z) and g(x, y, z) = k Evaluate f at all points (x, y, z) that result from the step above. The largest of these values is the maximum value of f; the smallest is the minimum value of f. 9
11 And there you go. You ll never have to search endlessly through pages in your textbook just to find the right formula ever again. Because staring at the same 3 pages and flipping back & forth for 15 minutes before you finally realize what you re looking for is in another lesson is just a total waste of time. You can print this out, put it in your notebook, and use it to get your homework done in less time. And hey. got a secret for you: There s more where this came from. If you need a little extra boost passing your math class or hitting your ideal GPA, I ve got loads of videos, cheat sheet notes, and practice quizzes to help you make sure you ace every pop quiz and exam this semester. And they re all available for a fraction of the cost of a tutor. (Really.) [Start Now & Boost Your Next Test Grade] 10
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