what is a derivative? what is tangent line? what does differentiable mean? how do they all differ?

Question

Please explain in simple, easy to understand terms for a stupid person like me.

Answer 1

> what is a derivative?

Imagine you have a graph showing two related quantities. For the sake of argument, lets say you're graphing the progress of a volcano that's growing bigger every day, and the graph shows how tall the volcano is at any given moment.

Your graph might be "steep" in some sections, and it might more or less "level out" in other sections. In places where the graph is steep, it means the volcano is growing pretty quickly. In places where the graph is more "level," it means the volcano was more or less stable during that time.

A "derivative" would be a second graph that measures the "steepness" of the first graph. During time intervals where the first graph is steep, the derivative would have a high value. During time intervals where the first graph is level, the derivitave would have a low value.

Where the first graph shows how _high_ the volcano is every day, the derivative graph shows how _fast_ the volcano is growing every day. You can think of the derivative graph as showing the "speed" of the volcano as it inches upward.

In general, if one graph shows the _value_ of some quantity, the derivative graph shows the _rate_ at which that quantity is _changing_. The derivative graph shows the "steepness" (or "slope") of the first graph at any given point.

> what is tangent line?

A tangent line is a graphical representation of the slope of a graph at some chosen point.

In general, the graph will be some kind of curvy line. If you take a ruler and lay it on the graph so that the straight edge touches the graph at just one point, so that the edge is "parallel" to the direction of the graph at that point, then you have a tangent line. The tangent line will be steep if you chose a point where the curve is steep; and level if you chose a point where the curve is level. A tangent line lets you assign a number to the steepness of the curve: the steepness is just the SLOPE of the tanglent line (that is: draw two points on the tangent line, then take the difference between their y-coordinates and divide by the difference in their x-coordinates. The result is the slope).

Remember that the derivative graph shows the steepness of the first graph. And that is why tangent lines are related to derivatives. In principle, you could draw a bunch of tangent lines (on the first graph), measure the slope of each, then create a second graph showing the values of all the slopes. That second graph would be exactly the derivative graph. (In real life, you use calculus techniques--not graphing--to calculate the derivative; but the graphs and tangent lines give you an illustration of what the calculus is doing.)

> what does differentiable mean?

In the examples I gave, we talked about graphs that are nice "curvy" lines. But you can also imaging graphs that are not like that. For example, say you measured the volcano's height only once per day (at noon), and just made a single dot on the graph every day. In that case, you could measure the _average_ rate of the volcano's growth between noon Tuesday and noon Wednesday; but the graph isn't complete enough for you to measure the exact rate of growth at, say, 9:30 PM on Tuesday night.

From the point of view of derivatives and tangent lines, there is (in this case) no real "curve" on the graph for you to draw a tangent line against, and no real "slope" or "steepness" on the graph. This means you can't really create a derivative graph.

A graph that is nice and curvy everywhere is called "differentiable," which is just a way of saying that you can make a derivative of it. (There's an exact mathematical definition of "nice and curvy," which we won't go into.) A graph that consists of just disconnected points, is not differentiable, because you can't make a derivative of it.

Many graphs are differentiable in some sections, but not differentiable in others. For example, if a graph is mostly curvy but has some sharp "kinks" or "corners" in certain spots, then the graph is differentiable everywhere _except_ where the "kinks" and "corners" are. The steepness on the left side of a "kink" may be (say) 1.2, but then the steepness on the right side of the kink may suddenly drop to 0.58. You can't say exactly what the steepness is AT the kink; so that spot in the graph is not differentiable. (To say it a different way, the derivative is "undefined" at that point.)

Answer 2

A derivative is the slope of a curve. If you plug in a number to the derivative, you will get the slope of the original function at that point. For example, y = x^2 so y' = 2x. The slope of x^2 at any point is 2x. So the slope of x^2 at 5 is 10.

A tangent line is a line that just touches the curve. The derivative is a tangent line. The tangent line to a curve takes the slope of the curve at the point of intersection.

The term differentiable is talking about functions that do not have a derivative at a point. A function that comes to a point such as functions involving absolute values are not differentiable at the point. So, if y = |x|, the the derivative of y in general is 0, but the derivative at the point (x=0) is undefined.

Hope this helps.

Answer 3

first derivative: 3x^2 - 4 evaluated at x= 2 12-4 = 8 The slope at x = 2 is 8. Lets see what point the tangent line has to go through by plugging in x=2 to original equation: y = 8 - 8 = 0 so we want the tangent line at (2,0) and the slope is 8. Use point-slope form of a line: y = 8(x-2) or y= 8x - 16