1. Chapter 2 Section 2 The Derivative!

2. Definition The derivative of a function f(x) at x = a is defined as f’(a) = lim f(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) = x 3 + x – 1 at x = 1 Start with f(1 + h) – f(1)/h

4. General Example! Find the derivative of f(x)=x 3 +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/dx f(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) = x 2 √(x 3 + 2) at x = 1.