1. Chapter 2 Section 2 The Derivative!

2. Definition • The derivative of a function f(x) at x = a is defined as f’(a) = limf(a+h) – f(a) • h->0 h • Given that a limit exists. • Then f is differentiable at x = a.

3. Example! • Find the derivative of f(x) = x3 + x – 1 at x = 1 • Start with f(1 + h) – f(1)/h

4. General Example! • Find the derivative of f(x)=x3+x-1 at some point x. (this point we don’t know)Differentiation • The derivative of f(x) to get the new function f’(x) given a limit exists. The process is called differentiation.

5. Derivative of a sqrt function • If f(x) = √x • What do the x’s have to be? • We need to figure out how to derive a new function from this using our formula.

6. Now to some graphing ?!? • Let’s look at some graphs of functions.

7. More graphing!!! • Graphs of derivatives.

8. Alternative notation • f’(x) = y’ = dy/dx = df/dx = d/dxf(x) • Where d/dx is called the differential operator • Or tells you to take the derivative of f(x)

9. Theorem 2.1 • If f(x) is differentiable at x = a then f(x) is continuous at x = a. • EXAMPLE TIME!!!!!!!!!!!!!!!!!

10. Show f(x) = 2 if x > 2 and 2x if x≥2 • At x = 2. • Let’s graph it! And then check our LIMITS!!!

11. Some non differentiable exampples • See Page 171, basically if there is a discontinuity in the graph, it is not differentiable at that point. • Or a “cusp” or “Vertical Tangent” line.

12. Approximating a derivative/velocity numerically • Use the function to evaluate the limit of the slopes of secant lines! • Use f(x) = x2√(x3 + 2) at x = 1.