How do limits connect to derivatives

How do derivatives apply to real life? How does a partial derivative differ from an ordinary derivative? How does calculus differ from algebra? How does calculus relate to biology? What is the derivative of x? What is the point of calculus? What is the formal definition of a derivative? You can understand the relation between a limit and a derivative if you look at their definition. In general, we are not interested in the function's behaviour at the point we are evaluating the limit.

As you can see a derivative is defined on top of the limit concept, so this is the relation between the two. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Ask Question. Asked 6 years, 8 months ago.

Active 1 year ago. Viewed 13k times. This rate of change is always considered with respect to change in the input variable, often at a particular fixed input value. In what follows, we will introduce terminology and notation that makes it easier to talk about the instantaneous rate of change of a function at a point. In addition, just as instantaneous velocity is defined in terms of average velocity, the more general instantaneous rate of change will be connected to the more general average rate of change.

Specifically, we make the following definition. For now, we make the following important notes. While all of the above ideas are important and we will add depth and perspective to them through additional time and study, for now it is most essential to recognize how the derivative of a function at a given value represents the slope of a certain line. Thus, we expand upon the last bullet item above.

Because this process of taking a limit is a dynamic one, it can be helpful to use computing technology to visualize what the limit is accomplishing. While there are many different options 3 , one of the best is a java applet in which the user is able to control the point that is moving.

See the examples referenced in the footnote here, or consider building your own, perhaps using the fantastic free program Geogebra 4. A secant line to a curve is simply a line that passes through two points that lie on the curve. In the situation where the limit of the slopes of the secant lines exists, we say that the resulting value is the slope of the tangent line to the curve.

There are scores of other examples posted by other authors on the internet. The following example demonstrates several key ideas involving the derivative of a function. Example 1.