1. MathematicalMarketing Slide 3a.1 Mathematical Tools Chapter 3: Mathematical Tools We will be reviewing  Exponents and Logarithms.  Scalar Calculus  Partial Derivatives of a Scalar Function of a Vector

2. MathematicalMarketing Slide 3a.2 Mathematical Tools The Ln Function  ln ab = ln a + ln b  ln (a/b) = ln a – ln b  ln a b = b ln a  ln e a = a  ln e = 1  ln 1 = 0 ln a = c if e c = a

3. MathematicalMarketing Slide 3a.3 Mathematical Tools Rules for Exponents a b · a c = a b+c a ½ = √a a -1 = 1/a We often say exp(a) instead of e a.

4. MathematicalMarketing Slide 3a.4 Mathematical Tools The Derivative Is the Limit of a Slope at a Point x x +  x x2x2 (x +  x) 2

5. MathematicalMarketing Slide 3a.5 Mathematical Tools Rise Over Run

6. MathematicalMarketing Slide 3a.6 Mathematical Tools Some Rules for Scalar Derivatives d(c)/dx = 0 f(x) = x m f(x) = c f(x) = cx f(x) = x 2 dx 2 /dx = 2x dcx/dx = c DerivativeFunction f(x) = exp(x)

7. MathematicalMarketing Slide 3a.7 Mathematical Tools Compound Functions The derivative of a sum The derivative of a function of a function The derivative of a function of an exponent

8. MathematicalMarketing Slide 3a.8 Mathematical Tools Partial Derivatives By definition A very simple example:

9. MathematicalMarketing Slide 3a.9 Mathematical Tools The Derivative of a Linear Combination

10. MathematicalMarketing Slide 3a.10 Mathematical Tools The Derivative of a Quadratic Form

11. MathematicalMarketing Slide 3a.11 Mathematical Tools The Transpose of a Derivative Is the Derivative of the Transpose

12. MathematicalMarketing Slide 3a.12 Mathematical Tools Second Order Derivatives