Partial derivatives BUSINESS MATHEMATICS 1
CONTENTS Derivatives for functions of two variables Higher-order partial derivatives Derivatives for functions of many variables Old exam question Further study 2
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Can we find the extreme values of a function g x, y of two variables x and y? Try g x, y = fails! g x, y = x 3 y + x 2 y 2 + x + y 2 3
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Can we find the extreme values of a function g x, y of two variables x and y? g x, y = x 3 y + x 2 y 2 + x + y 2 Try g x, y = fails! Recall definition of derivative of a function f x of one variable: f x = df x dx = lim f x + h f x h 0 h Generalization to partial derivative of g x, y of 2 variables: g x, y = lim h 0 g x + h, y g x, y h 4
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Differentiate g x, y = x 3 y + x 2 y 2 + x + y 2 with respect to x (and keeping y fixed) g = 3x2 y + 2xy 2 + 1 5
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Differentiate g x, y = x 3 y + x 2 y 2 + x + y 2 with respect to x (and keeping y fixed) g = 3x2 y + 2xy 2 + 1 and with respect to y (and keeping x fixed) g y = x3 + 2x 2 y + 2y 6
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Differentiate g x, y = x 3 y + x 2 y 2 + x + y 2 with respect to x (and keeping y fixed) g = 3x2 y + 2xy 2 + 1 and with respect to y (and keeping x fixed) g y = x3 + 2x 2 y + 2y Clearly, in this case g g y. Therefore, never write f for a function of two variables! 7
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES So we define the partial derivative of f with respect to x f x, y = lim h 0 f x + h, y f x, y h and similar with respect to y f x, y y = lim h 0 f x, y + h f x, y h 8
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES f x, y f x, y y 9
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Alternative notations f, f x,y, f x, f 1, f x, f 1, x f, Not important to remember, but important to recognize so, basically a lot of choice, but never write df or f dx 10
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES The partial derivative in a point is a number: f x,y x,y = 2, 5 = 3 The partial derivative over a range of points is a function of x and y: f x,y = 2x + 3y 6 11
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Example: Cobb-Douglas production function describing how a firm s output q depends on capital input (K) and labour input (L): q K, L = A K α L β where A, α, and β are positive constants. 12
DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Example: Cobb-Douglas production function describing how a firm s output q depends on capital input (K) and labour input (L): q K, L = A K α L β where A, α, and β are positive constants. Marginal productivity of capital q K = A α Kα 1 L β Note that when 0 < α < 1, q K diminishing marginal returns. is a decreasing function of K leading to 13
EXERCISE 1 Given is f x, y = x y. Find f and f y. 14
HIGHER-ORDER PARTIAL DERIVATIVES Recall the second derivative d dx df x dx = d2 f x dx 2 = f x Four possibilities for function g x, y : y y g g y g g y = 2 g 2 = 2 g y 2 = 2 g y = 2 g y 16
HIGHER-ORDER PARTIAL DERIVATIVES Recall the second derivative d dx df x dx = d2 f x dx 2 = f x Four possibilities for function g x, y : y y g g y g g y = 2 g 2 = 2 g y 2 = 2 g y = 2 g y so, never d2 g dx2 or g Alternative notations: 2 g g x,y, y y, g yx, g 21, g yx, g 21, xy g,. 17
HIGHER-ORDER PARTIAL DERIVATIVES Example: g x, y = x 3 y + x 2 y 2 + x + y 2 2 g 2 = 6xy + 2y2 2 g = y 2 2x2 + 2 2 g = y 3x2 + 4xy 2 g = y 3x2 + 4xy 18
HIGHER-ORDER PARTIAL DERIVATIVES Example: g x, y = x 3 y + x 2 y 2 + x + y 2 2 g 2 = 6xy + 2y2 2 g = y 2 2x2 + 2 2 g = y 3x2 + 4xy 2 g = y 3x2 + 4xy For almost all functions 2 g y = 2 g y and certainly for all functions we encounter in business and economics. 19
EXERCISE 2 Given is f x, y = 4x 3 y 2 3y 4 e 2x. Find 2 f y in x, y = 1,0. 20
HIGHER-ORDER PARTIAL DERIVATIVES Likewise, we can define third-order derivatives f x,y = 3 f 3 y y f x,y = 3 f y 2 How many are there? How many are different? 22
HIGHER-ORDER PARTIAL DERIVATIVES Likewise, we can define third-order derivatives f x,y = 3 f 3 y y f x,y = 3 f y 2 How many are there? How many are different? And even higher-order partial derivatives n f, n n f n 1 y,. 23
DERIVATIVES FOR FUNCTIONS OF MANY VARIABLES For functions f x 1, x 2, x 3,, x n partial derivatives we can form n first-order f, f,, 1 2 f n and many many second-order partial derivatives 24
EXERCISE 3 Given is g x = 1 σ n i=1 n x i. Find g 4. 25
OLD EXAM QUESTION 27 March 2015, Q1b 27
OLD EXAM QUESTION 22 October 2014, Q1h 28
FURTHER STUDY Sydsæter et al. 5/E 11.1-11.2 Tutorial exercises week 2 partial derivatives higher-order partial derivatives partial derivatives graphically 29