1. Derivatives and the Tangent Line

2. Blast from the past…

3. The New vs. The Old
Pre-Calculus concept: Slope of a line.
This is a static or constant concept.
Uses a formula.
Calculus concept: Slope of a curve at a given point.
This is a variable concept.
Uses formulas.

4. The Process…In Theory
Theoretically, we are “drawing” a secant line through that point and a nearby point (Δy/Δx).
We then "slide" the second point along the curve toward the first (remember…math in motion!);
The secant line gets closer and closer to being a tangent line.

5. The Process…In Theory
f(x+x)
f(x) x
x+x

6. The Process…In Theory x
In the limit, as the distance between the two points goes to 0, the secant line becomes the tangent line. x

7. Side note: Does the formula for this limit look vaguely familiar?
What if the x were changed to an h?
Difference Quotient (version 2.0):

8. The Process…In Theory
Its slope becomes the slope of the curve at that point, called the derivative.
Expressed: x

9. Symbols for Derivative

10. Sample Problem: Find the derivative of f(x)=x2+4x+4   
Picture from

11. The Process

12. The Process

13. Try This One On Your Own
Calculate the derivative of

14. Sample Problem Find the slope of the tangent to the curve
at the point (6,4).

15. The Process

16. The Process