Understanding Mathematical Tools: Exponents, Logarithms, and Scalar Calculus

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

2. The Ln Function • ln ab = ln a + ln b • ln (a/b) = ln a – ln b • ln ab = b ln a • ln ea = a • ln e = 1 • ln 1 = 0 ln a = c if ec = a

3. Rules for Exponents ab· ac = ab+c a½ = √a a-1 = 1/a We often say exp(a) instead of ea.

4. The Derivative Is the Limit of a Slope at a Point (x +  x)2 x2 x x +  x

5. Rise Over Run

6. Some Rules for Scalar Derivatives Function Derivative f(x) = c d(c)/dx = 0 f(x) = cx dcx/dx = c f(x) = x2 dx2/dx = 2x f(x) = xm f(x) = exp(x)

7. Compound Functions The derivative of a sum The derivative of a function of a function The derivative of a function of an exponent

8. Partial Derivatives By definition A very simple example:

9. The Derivative of a Linear Combination

10. The Derivative of a Quadratic Form

11. The Transpose of a Derivative Is the Derivative of the Transpose

12. Second Order Derivatives