Chapter 2 Section 2The Derivative!

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.

Example!

Find the derivative of f(x) = x3 + x – 1 at x = 1

Start with f(1 + h) – f(1)/h

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.

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.

Now to some graphing ?!?

Let’s look at some graphs of functions.

More graphing!!!

Graphs of derivatives.

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)

Theorem 2.1

If f(x) is differentiable at x = a then f(x) is continuous at x = a.

EXAMPLE TIME!!!!!!!!!!!!!!!!!

Show f(x) = 2 if x > 2 and 2x if x≥2

At x = 2.

Let’s graph it! And then check our LIMITS!!!

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.

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.