1. 2.1 The Derivative and The Tangent Line Problem
The Definition of a Derivative

2. WARM UP: Find the slope of the tangent line
to the curve at the given point

3. The DERIVATIVE finds the slope of the tangent
line to a given function at a given point.
Know the different notations:

4. Comparison of SLOPE and DERIVATIVE:
Slope between 2 points: Average “Rate of Change”
DERIVATIVE:
Slope at 1 point Instantaneous Rate of Change

5. Places where there is NO derivative:
Discontinuity
Vertical Tangent
Cusp

6. Theorem: If f(x) is differentiable,
it IS continuous.
***BUT, all continuous functions are
NOT differentiable.

7. Alternate form of derivative to the graph
at x=c:
As long as the one-sided limits from the right and
from the lift exist and are equal,

8. Ex) Find the derivative by the limit process.
Plan:
Find the slope of the tangent line to the graph at any point (x,f(x))

9. Ex) Describe the x-values at which f is
differentiable.
Plan:
Omit parts of graph where there are discontinuities, vertical asymptotes and cusps.

10. Ex) Describe the x-values at which f is
differentiable.
Plan:
Omit parts of graph where there are discontinuities, vertical asymptotes and cusps.

11. Ex) Describe the x-values at which f is
differentiable.
Plan:
Omit parts of graph where there are discontinuities, vertical asymptotes and cusps.

12. Ex) Use the alternate form of derivative to
find the derivative at x=c, if it exists.
Plan:
Use the alternate form of derivative…check that limit exists.