The Derivative

Today, we started our discussion on the derivative. The derivative is a function for the slope of the tangent line everywhere. We found the derivative using the limit definition that can be found on page 95. We found that if f(x) = x^2, then f'(x) = 2x. This means that the slope of the tangent line for f(x) = x^2 at x=3 is
f'(3) = 2(3) = 6.

We also spent some time looking at the graph of f'(x) as it related to f(x). For right now, we are only approximating the slopes of the given graph. Mostly, we look to see when the given graph has positive, negative, or zero slopes to help us determine the derivative graph. Soon, we will look at graphing the derivative of the function with a little more precision.