1. Basic Differentiation Rules

2. Derivative Rules
Theorem. [The Constant Rule] If k is a real number such that for all x in some open interval I, then for all
Theorem. [The Power Rule] Let r be a rational number, and let Then
for all values of x where this expression is defined.

3. Examples Find derivatives for the following functions:
Find the equation of the line tangent to the graph of at the point

4. More Derivative Rules
Theorem [The Constant Multiple Rule] Let k represent a real number, and let f be a differentiable function. Then the function kf is also differentiable and
Example. Find the derivative of

5. Theorem [The Sum and Difference Rules] Let f and g be differentiable functions. Then
Example. Find the derivative of each function.
Note. This theorem generalizes to any finite sum or difference.

6. Theorem.
Example. Find all values of x where the line tangent to the graph of has slope –1.

7. The Derivative As a Rate of Change
Slope.

8. Velocity. Let be a function giving the position of a point moving on a number line at time t.
The derivative gives the instantaneous velocity at time

9. The Derivative an Instantaneous Rate of Change

10. Example. A stone dropped from a bridge falls in t seconds
Example. A stone dropped from a bridge falls in t seconds. Find the velocity after 3 seconds. If a river flows 256 feet below the bridge, with what velocity does the rock enter the water?