1. Derivative and the Tangent Line Problem
The beginnings of Calculus

2. Tangent Line Problem
Definition of Tangent to a Curve Now to develop the equation of a line we must first find slope

3. Definition of Tangent Line with Slope m
Slope of Secant Line
If f is defined on an open interval containing c, and if the limit
exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)).

4. Definition of Tangent Line
The limit is telling us that the distance between the two points is getting smaller and smaller. If we let the limit approach zero, we are actually approaching being on one point and not two. That is how we can say the line is tangent and no longer a secant line.

5. Definition of Tangent Line
Find some slopes using this definition.

6. Definition of the Derivative of a Function
Demonstration

7. Symbols for Derivative
Sir Isaac Newton:
f’(x) or y’
Gottfried Leibniz:
This is read as “the derivative of y with respect to x.”

8. Finding the Derivative by the Limit Process
Examples:
Quadratic Equation
Square Root Function
Rational Function

9. Finding the Derivative Summation
A very good summation of this information

10. Differentiation Definition
Alternative form of derivative: (This is when given one point)

11. When is a Function not Differentiable
When the graph has a sharp point. (This is because the derivative (slope) from the right and the derivative from the left are different values.

12. When is a Function not Differentiable
When the graph has a vertical line tangent. Remember the slope of a vertical line is undefined.

13. When is a Function not Differentiable
Where the function is not continuous at that point.

14. Differentiability Implies Continuity
If f is differentiable at x = c, then f is continuous at x = c.
The converse is not true. Just because a function is continuous does not make it differentiable everywhere. (See the prior slides)