The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

The Physical Concept of the Derivative

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

$$\mbox{Average velocity} = \frac{\mbox{distance from A to B }}{\mbox{time to get from A to B }}$$

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

$$\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot$$

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have

$$\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot$$

If f(t) describes the position at time t, then $\Delta s = f(a + \Delta t) - f(a)$. In this case, we have

$$\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{f(a + \Delta t) - f(a)}{\Delta t}\cdot$$

Example. Consider a parabolic motion given by the function f(t) = t². The instantaneous velocity at t=a is given by

$$\lim_{\Delta t \rightarrow 0} \frac{f(a + \Delta t) - f(a)}{\... ...a t \rightarrow 0} \frac{(a + \Delta t)^2 - a^2}{\Delta t}\cdot$$

Since

$$\frac{(a + \Delta t)^2 - a^2}{\Delta t} = \frac{2 a \Delta t + \Delta t^2}{\Delta t} = 2a + \Delta t,$$

we conclude that the instantaneous velocity at t=a is 2 a.

This concept of velocity may be extended to find the rate of change of any variable with respect to any other variable. For example, the volume of a gas depends on the temperature of the gas. So in this case, the variables are V (for volume) as a function of T (the temperature). In general, if we have y = f(x), then the average rate of change of y with respect to x from x = a to $x = a + \Delta x$, where $\Delta x \neq 0$, is

$$\mbox{Average Rate} = \frac{\Delta y}{\Delta x} = \frac{f(a + \Delta x) - f(a)}{\Delta x}\cdot$$

As before, the instantaneous rate of change of y with respect to x at x = a, is

$$\mbox{Instantaneous Velocity (at $x=a$)} = \lim_{\Delta x \ri... ...a x \rightarrow 0} \frac{f(a + \Delta x) - f(a)}{\Delta x}\cdot$$

Notation. Now we get to the hardest part. Since we can not keep on writing "Instantaneous Velocity" while doing computations, we need to come up with a suitable notation for it. If we write dx for $\Delta x$ small, then we can use the notation

$$\mbox{Instantaneous Velocity (at $x=a$)} = \frac{d y}{dx}(a)\cdot$$

This is the notation introduced by Leibniz. (Wilhelm Gottfried Leibniz (1646-1716) and Isaac Newton (1642-1727) are considered the inventors of Calculus.)

The Geometrical Concept of the Derivative

Consider a function y = f(x) and its graph. Recall that the graph of a function is a set of points (that is (x,f(x)) for x's from the domain of the function f). We may draw the graph in a plane with a horizontal axis (usually called the x-axis) and a vertical axis (usually called the y-axis).

Fix a point on the graph, say (x₀, f(x₀)). If the graph as a geometric figure is "nice" (i.e. smooth) around this point, it is natural to ask whether one can find the equation of the straight line "touching" the graph at that point. Such a straight line is called the tangent line at the point in question. The concept of tangent may be viewed in a more general framework.

(Note that the tangent line may not exist. We will discuss this case later on.) One way to find the tangent line is to consider points (x,f(x)) on the graph, where x is very close to x₀. Then draw the straight-line joining both points (see the picture below):

As you can see, when x get closer and closer to x₀, the lines get closer and closer to the tangent line. Since all these lines pass through the point (x₀,f(x₀)), their equations will be determined by finding their slope: The slope of the line passing through the points (x₀,f(x₀)) and (x,f(x)) (where $x \neq x_0$) is given by

$$m(x) = \frac{f(x) - f(x_0)}{x-x_0}\cdot$$

The tangent itself will have a slope m, which is very close to m(x) when x itself is very close to x₀. This is the concept of limit once again!

In other words, we have

$$m = \lim_{x \rightarrow x_0} m(x) = \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x-x_0}\cdot$$

So the equation of the tangent line is

$$y - f(x_0) = m (x-x_0)\cdot$$

Notation. Writing "m" for the slope of the tangent line does not carry enough information; we want to keep track of the function f(x) and the point x₀ in our notation. The common notation used is

m = f'(x₀).

In this case, the equation of the tangent line becomes

y - f(x₀) = f'(x₀) (x-x₀)

where

$$f'(x_0) = \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x-x_0}\cdot$$

One last remark: Sometimes it is more convenient to compute limits when the variable approaches 0. One way to do that is to make a translation along the x-axis. Indeed, if we set h=x-x₀, we get

$$f^\prime(x_0)=\lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x-x_0} = \lim_{h \rightarrow 0} \frac{f(x_0 + h) - f(x_0)}{h}\cdot$$

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