1. The Derivative
Chapter 3.1 Continued

2. The Derivative
The slope of a tangent line to y = f(x) at the point where x = a is called the derivative of the function at a. It is written a number of different ways:

3. Let’s use the definition of the derivative at a point to find the slopes of tangent lines to y = x at the points where x = 1, ½, 0, -1 x
slope = f’(x) 1 -1 x
f (x) 1 2
1.25 -1

4. The Derivative as a Function
What’s the Rule? The derivative is 2 times each x-coordinate.
With f(x) = x2 + 1, x
slope = f’(x) 1 2 -1 -2

5. A Function and its Derivative
f (x) = x f’(x) = 2x

6. Instead of calculating the slope separately for each value of x, we can find the derivative in general and then, if desired, plug in particular values of x.
We define the derivative of a function as follows:

7. Examples
Let’s use the definition to differentiate the following functions. 1. f(x) = 8x – 7 2. f(x) = 3. f(x) = x2 + 3x + 2

8. Voting Question

9. Differentiability
A function must be continuous at a point in order to have a derivative there, but that’s not enough. A continuous function that has a sharp corner or vertical tangent line at a point does not have a derivative at that point. If a graph contains a sharp turn or a vertical tangent line, we say the derivative is undefined at that point, and the function is not differentiable there.

10. Points of non-differentiability

11. When is a Function not Differentiable at a Point?
A function f is not differentiable at a if at least one of
the following conditions holds:
f is not continuous at a
f has a corner at a
f has a vertical tangent at a

12. Let’s Vote
At each labeled point, is the derivative positive, negative, zero, or undefined?
Positive
Negative
Zero
Undefined

13. Graphing the Derivative of a Function
The derivative is the slope of the original function.
The derivative is defined
at the end points of a
function on a closed
interval.