In this lesson, our instructor Vincent Selhorst-Jones goes over Instantaneous Slope & Tangents (Derivatives). Youll review the definition of a derivative and slope, before exploring the idea of instantaneous slope. Vincent also teaches about the tangent of a curve and the ways to look at average slope, the general form, and limits. You learn about notations of derivatives and the Power Rule before diving into three onscreen practice examples.
Long ago in algebra, we were introduced to the concept of slope. We can think of slope as the rate of change that a line has. We define slope as the vertical change divided by the horizontal:
The idea of slope makes sense for a line, because the slope is always the same, no matter where you look. However, on most functions, the slope is constantly changing. While we can't talk about a rate of change/slope for the entire function, we can look
for a way to find the instantaneous slope: the slope at a point.
Another way to consider the idea of instantaneous slope is through a tangent line: a line going in the "same direction" as a curve at some point, and just touching that one point.
We can find the average slope between two points on a function with
f(x2) − f(x1)
x2 − x1
Alternatively, we can rephrase the above formula by considering the average slope between some location x and some location h distance ahead:
f(x+h) − f(x)
Notice that as h grows smaller and smaller, our slope approximation becomes better and better. With this idea in mind, we take the limit of the above as h→ 0. Assuming the limit exists, we have the derivative:
f(x+h) − f(x)
[We read `f′(x)' as `f prime of x'.] By plugging a location into f′(x), we can find the instantaneous slope at that location. We call the process of taking a derivative differentiation.
There are many ways to denote the derivative. Given some y = f(x), we can denote it with any of the following:
The first two are the most common, though.
As you progress in calculus, you will quickly learn a wide variety of rules and techniques to make it much easier to find derivatives. You will very seldom, if ever, use the formal definition to find a derivative. The really important idea is what
a derivative represents. A derivative is a way to talk about the instantaneous slope (equivalently, rate of change) of a function at some location. No matter how many rules you learn for finding derivatives, never forget that it is, at heart, a way to
talk about a function's moment-by-moment change.
One rule to make it easier to find the derivative of a function is the power rule. It says that for any function f(x) = xn, where n is a constant number, the derivative is
f′(x) = n ·xn−1.
Instantaneous Slope & Tangents (Derivatives)
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.