The document provides an introduction to formal derivatives, highlighting its definition and the process of computation. It includes examples of deriving functions such as f(x) = 7x + 3 and f(x) = x² + 7x + 3, demonstrating the steps involved in formal derivation. Additionally, it poses questions regarding changes in computation for different types of functions, including higher-degree polynomials and trigonometric functions.

1. Formal Derivatives A Short Orientation with Examples By Gideon L. Weinstein Course Mentor Western Governors University Last updated May 2010

2. Definition of the Derivative The formal definition of the derivative f’(x) of a function f(x) is

3. Key Point Please note that in this definition, x is a constant and it is h that we treat as the variable as it approaches zero The key to these computations is to do as much algebra as possible BEFORE taking the limit.

4. Example 1 Let f(x) = 7x + 3 and formally compute f’(x). Please note that in a formal proof, there would be a justification for each the steps taken. That is left as an exercise for the reader.

5. Example 1, continued So when f(x) = 7x + 3 then f’(x) = 7.

6. Example 2 Let f(x) = x 2 + 7x + 3 and formally compute f’(x).

7. Example 2, continued So when f(x) = x 2 + 7x + 3 then f’(x) = 2x + 7.

8. Extensions How would the computation change if the first example was 3x + 7 instead? How would the computation change if the second example was 5x 2 + 7x + 3 instead? What happens with higher-degree polynomials? What happens with functions such as sines, cosines, logarithms, and exponentials?