1. Tangent Line Recall from geometry
Tangent is a line that touches the circle at only one point
Let us generalize the concept to functions
A tangent will just "touch" the line but not pass through it
Which of the above lines are tangent?

2. A Secant Line Crosses the curve twice The slope of the secant will be
x = a h

3. Tangent Line Now let h get smaller and smaller
x = a h
The slope of the tangent line

4. The Derivative We will define the derivative of f(x) as Note
The derivative is the rate of change function for f(x)
The derivative is also a function of x
The limit must exist

5. Comparison Difference Quotient Slope of secant Average rate of change
Average velocity
Derivative
Slope of tangent
Instantaneous rate of change
Instantaneous velocity

6. “the derivative of f with respect to x” (LaGrange’s Notation)
“f prime of x” or
“the derivative of f with respect to x”
(LaGrange’s Notation)
“y prime” or
“the derivative of y with respect to x”
(Leibniz’s Notation) or
“the derivative of f with respect to x” or
“the derivative of f of x”

8. Tangent Line Given Evaluate Once you have the slope and the point
Determine the slope of the tangent line at x = 0
Evaluate
Once you have the slope and the point
You can determine the equation of the line

10. Examples:
Find the derivative of the following functions 1) 2) 3)

11. Try It Out Use the strategy to find the derivatives of these functions. ,

12. Equation of the Tangent Line
We stated previously that once we determine the slope of the tangent

13. Tangent Line
Write equations of tangent lines thru x = 1 for each function:

14. Warning
Our definition of derivative included the phrase "if the limit exists"
Derivatives do not exist at "corners" or "sharp points" on the graph
The slope is different on each side of the point
The limit does not exist
f(x) = | x – 3 |