Review differentiation.

What are derivatives?

After working with limits, our main tool last semester was the derivative. We started with a function and a number in the interior of its domain. The derivative of at ( denoted as either or ) can be interpreted in a couple different ways. It represents the instantaneous rate of change of at . Graphically it represents the slope of the line tangent to the graph of at the point .

When we first came up with the idea, we didn’t know how to calculate explicitly, so we approximated instead. We approximated the instantaneous rate of change at by the average rate of change on . Graphically we approximated the slope of the tangent line at by the slope of the secant line through and . The exact value arose when we took the limit as tended to .

The definition of the derivative is then:

Let’s go through an example of using this definition.

Shortcut formulas

We soon turned to finding ‘shortcut formulas’ for finding derivatives more easily. The most important of these are below.

Power Rule
Sum Rule
Product Rule
Quotient Rule
Chain Rule
Exponential and Logarithmic Functions
Trigonometric Functions

Applications

We finished last semester with a few applications that showed some of the things the derivative was good for. (There are MANY others!)

Recall that a critical point of a function is a number with either or does not exist.

Find the critical points of the function .

Critical points played a large role for us in graphing functions and in optimization problems. Try the following optimization problem.

Among all rectangles with perimeter , find the maximum area possible.