Course:MATH110/Archive/2010-2011/003/Notes/What can we do with derivatives

It is a good thing to be able to compute derivatives, but what for? Here's a running list of things derivatives can do for us. It is recommended that you keep a similar list for yourself and add more details to it as the course moves along.

• Derivatives allow us to compute the equation of the tangent line to a function at a given point.

The derivative tells us about the slope of that tangent line and we are able to get the coordinates of the point where the tangent line is taken (just compute the second coordinate, you usually have the first one as a given). This is enough to compute the equation of that tangent line.

• The slope of the tangent line is the rate of change of the function. For example, if a function gives us the distance that is being covered as a function of time, the derivative is the instantaneous speed.

• The derivative of a function gives you a measure of its growth. Knowing when the derivative of a function is positive or negative will tell you when the original function is increasing or decreasing respectively.

• We can compute higher order derivatives. Once we're given a function, we can compute its derivative and then take the derivative again to get the second derivative, and so on! We'll see later how to use the second derivative to get some valuable information on the initial function.

• Derivatives are a key ingredient for optimization. This topic will be covered later in the course. In the meantime, you're encouraged to check out the wikipedia article on optimization for example.