1. 1. Definition of Derivative

2. Calculus overview
3 main concepts
Limits
Derivatives
Integrals

3. What is the derivative?
The derivative is the slope of the tangent line at a point.
Therefore if you have a graph, you can estimate the derivative by looking at the slope of the graph at that point

4. Example 1
Estimate the derivative at the points marked on the graph below.

5. Slope of secant line vs. tangent line
As h shrinks to zero, slope of secant line approaches slope of tangent line
Slope of tangent line =

6. Derivative
The formula for slope of a tangent line (with the limit) is called the derivative
Can also be written
This will give a numeric value

7. Secant vs. tangent Secant Line Tangent Line
Uses 2 points (a, f(a)) and (b,f(b))
Uses 1 point (c, f(c))
The following are equivalent:
Slope of secant line between [a,b]
Slope of tangent line at x = c
Average rate of change on [a,b]
Instantaneous rate of change at
x=c
f’(c)
Derivative at x = c
dy/dx

8. Example 2
For f(x) = 4x2 , use the definition of the derivative to find the derivative function f’(x), then find the equation of the tangent line to f(x) at x=2.

9. Example 3
For g(t) = t/(t+1), use the definition of the derivative to find the derivative function g’(t), then find the equation of the normal line to g(t) at t=1.

10. Example 4
Find the derivative using the alternative from of the definition of f(x)= x2 – 3x at x=1

11. Differentiability
A function f(x) is differentiable at x=c if and only if
L is a finite value
This basically says that for a function to be differentiable at a point, the graph must be continuous but also must connect in a way that the slopes merge into each other smoothly.

12. Example 5
Discuss the differentiability of
at x = 2

13. Example 6
Discuss the differentiability of
at x = 0

14. Differentiability Places a function is not differentiable
Anywhere there is a discontinuity
Corner or cusp
Vertical tangent

15. Local Linearity
A function that is differentiable closely resembles its own tangent line when viewed very closely
In other words, when zoomed in on a few times, a curve will look like a straight line if it is differentiable.

16. Example 7

17. Continuity vs. Differentiability
If f(x) is differentiable at x = c, then f is continuous at x = c.
Be careful! The converse is not true! Being continuous does not imply differentiability (think about the absolute value function)

18. Piecewise function
You must always be careful about piecewise functions
Example 8
For the piecewise function find a piecewise derivative function f’(x) by using the definition of the derivative. Then evaluate the following
What can you say about the differentiability based on your results?