Differentiation Rules of Differentiation Differentiation for a Function

Differentiation

Rules of Differentiation • Differentiation for a Function of One Variable • Differentiation Involving Two or More Functions of the Same Variable • Differentiation Involving Functions of Different Variables • Partial Differentiation • Constant-Function Rule • Power-Function Rule Generalized 2

A Review of the 9 Rules of Differentiation for a Function of One Variable 3

Constant-Function Rule 4

Power-Function Rule 5

Rules of Differentiation Involving Two or More Functions of the Same Variable • Sum-difference rule • Product rule • Finding marginal-revenue function from average-revenue function • Quotient rule • Relationship between marginal-cost and average-cost functions 6

Sum or difference rule 7

Product rule 8

Finding marginal-revenue function from average-revenue function using the product rule 9

A Review of the 9 Rules of Differentiation for a Function of One Variable 10

Quotient rule 11

Relationship between marginal-cost and average-cost functions C MC AC Q 12

Rules of Differentiation Involving Functions of Different Variables • Chain rule • Inverse-function rule 13

A Review of the 9 Rules of Differentiation Involving Functions of Different Variables 14

Chain rule 15

Chain rule 16

Chain rule and its relation to total differential 17

A Review of the 9 Rules of Differentiation Involving Functions of Different Variables 18

Inverse-function rule 19

Inverse-function rule 20

Inverse-function rule • This property of one to one mapping is unique to the class of functions known as monotonic functions: • Recall the definition of a function, p. 17, function: one y for each x monotonic function: one x for each y (inverse function) • if x 1 > x 2 f(x 1) > f(x 2) monotonically increasing Qs = b 0 + b 1 P supply function (where b 1 > 0) P = -b 0/b 1 + (1/b 1)Qs inverse supply function • if x 1 > x 2 f(x 1) < f(x 2) monotonically decreasing Qd = a 0 - a 1 P demand function (where a 1 > 0) P = a 0/a 1 - (1/a 1)Qd inverse demand function 21

Partial Differentiation • Partial derivatives • Techniques of partial differentiation • Geometric interpretation of partial derivatives 22

Partial derivatives 23

Partial derivatives 24