4.2 Derivatives

The derivative refers to the slope of the tangent line at any point of a function.

Investigate the derivative of at the points (1,1), (2,4), (-3, 9).

Investigate the derivative of at the points (1, 1), (-2, -8), (3, 27).

Investigate the derivative of at the points (1, 1),

The investigations tell us that the derivative of a function in the form will be in the form .

In order to discuss derivatives algebraically we need to introduce the idea of a limit.

Last class we discussed the concept of a tangent being approximated by two points gradually getting closer together. (secant becoming a tangent) The notation we use looks like:

reads “the limit as x approaches a of “

Ex 1 Limits are a process to analyze a function, so let's analyze at x = 1

even though the denominator will equal 0 at x = 1, the limit only cares about the trend toward the value, not the value itself.

The derivative itself is a function. A function where the y-value is equal to the slope of the tangent to the original function at the value of x. We use limits a slightly different way to calculate a derivative function.

this limit takes any point on a function and examines what happens with very small changes to the x-value. This method of calculating a derivative is called first principles. The notation we use is , which reads “dee y dee x” and means “the derivative of y with respect to x”. is not a fraction, it is a single mathematical expression. Another way of expressing a derivative is or , which reads “y prime” or “f prime of x”.