## Math for Microeconomics

### Introduction

In this section we shall present the basic math tool to solve one variable optimization problems. In this course math is a tool to solve economics problems.
### Derivative

**Function $$****f**: For any element $$**x**, $$** f(x)** is an element in the range of $$**f** corresponding to element $$**x** in the domain. A function $$**f** can be represented as a graph $$**{(x,y): y =f(x)}** for all $$**x** in the domain of $$**f**.
The most common functions used in Economics are linear functions, functions to fractional exponents, and ln functions.

Linear: $$**f(x) = a + bx **

Fractional exponent: $$**f(x) = x**^{a/b}

Ln $$**f(x) = a + b ln(x) **

**Derivative $$****f'(x)**: The derivative of $$**f** at $$**x**, denoted as $$**f'(x)** or $$**df/dx** is defined as:

$$**f'(x) = lim([f(x+h) - f(x)]/h) as h -> 0 **

The type of functions we will consider in this course will be differentiable at almost all points in their domains.

It is important to note that the derivative has a geometric interpretation. The derivative $$**f'(x)** is the slope of the tangent line to $$**f** at $$**x**.

In taking derivatives we could go back to first principles and use the definition to derive them. It is much easier to use the rules derived from the definitions.

#### Derivative Rules

$$**1. f(x) = x**^{n} f'(x) = n x^{n-1}

$$**2. f(x) = x**^{a/b} f'(x) = (a/b) x^{(a/b)-1}

$$**3. f(x) = ln x f'(x) = 1/x<**
Examples:

$$**1. f(x) = x f'(x) = x**^{0} = 1

$$**2. f(x) = x**^{4} f'(x) = 4x^{3}

$$**3. f(x) = x**^{2/3} f'(x) = (2/3)x^{-(1/3)} = 2/[3x^{(1/3)}]

There are three rules involving functions of functions which we will use repeatedly in the course.

1 Product Rule: Given functions $$**g(x)** and $$**h(x)** with $$**f(x) = g(x)h(x)**:

$$**f'(x) = g'(x)h(x) + g(x)h'(x)**
2.Quotient Rule: Given functions $$**g(x)** and $$**h(x)** with $$**f(x) = g(x)/h(x)**:

$$**f'(x) = [g'(x)h(x) - g(x)h'(x)]/[h(x)]**^{2}
3.Composite Function Rule: Given functions $$**g(x)** and $$** y= h(x)** with $$**f(x) = g(h(x)) = g(y)** What this means is that for a given $$**x, y = h(x)** is the input for $$**g( )** :

$$**f'(x) = g'(y)h'(x)**

We could also write it as:

$$**df/dx = (dg/dy)(dy/dx)**.

Examples:
$$**1. f(x) = ln (x**^{2})

_$$** f'(x) = (1/x**^{2})2x = 2/x

$$**2. f(x) = g(x)h(x) = (x**^{2} - 3x)(2x^{3} - 1)

_$$** f'(x) = g'(x)h(x) + h'(x)g(x)**

_____$$** = (2x - 3)(2x**^{3} - 1) + (6x^{2})(x^{2} - 3x)

_____$$** = 10x**^{4} - 24x^{3} - 2x + 3

$$**3. f(x) = (3x**^{2} + 2x + 1)^{2}

_$$** f'(x) = 2(3x**^{2} + 2x + 1)(6x + 2)

$$**4. f(x) = g(x)/h(x) = x**^{2}/(2x - 1)

_$$** f'(x) = (g'(x)h(x) - h'(x)g(x))/(h(x))**^{2}

_____$$** = (2x(2x - 1) - 2x**^{2})/(2x - 1)^{2}

_____$$** = (2x**^{2} - 2x)/(2x - 1)^{2}

$$**5. f(x) = g(x)/h(x) = (x**^{2} - 3)/x^{2}

_$$** f'(x) = (g'(x)h(x) - h'(x)g(x))/(h(x))**^{2}

_____$$** = (2x(x**^{2}) - 2x(x^{2} - 3))/(x^{2})^{2}
= 6/x^{3}

$$**6. f(x) = g(x)h(x) = 2x**^{2}(3x^{4} - 2)

_$$** f'(x) = g'(x)h(x) + h'(x)g(x) = 4x(3x**^{4} - 2) + 2x^{2}(12x^{3})

_____$$** = 36x**^{5} - 8x

$$**7. f(x) = 3 ln(x**^{2} - 1)^{3}

_$$** f(x) = 3 ln(y);____ y = w**^{3}; ____w = x^{2} - 1

_$$** f'(x) = (3/y)3w**^{2}(2x)

_____$$** = (3 [3(x**^{2} - 1)^{2} (2x)]/(x^{2} - 1)^{3}

_____$$** = 18x/(x**^{2} - 1)

$$**8. f(x) = (x**^{2} + 1)^{3}(x^{3} - 1)^{2}

__$$** f(x) = uv;____ u = w**^{3}; w = x^{2} + 1; v = z^{2}; z = x^{3} - 1.

_** f'(x) = v (3w**^{2}2x) + u(2z3x^{2})

_____$$** = 6(x**^{3} - 1)^{2}(x^{2} + 1)^{2}x + 6(x^{2} + 1)^{3}(x^{3} - 1)x^{2}

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