Lesson 2.1: The Tangent Problem and Its Solution: the Derivative

Often we can find useful information from the slope of a tangent line to a graph. If a phenomenon is described by a complicated function, we can gather information about a small section of the function’s graph much more simply by examining the tangent to a point in that section.

Those of you who have taken physics will recall that if we have a graph of position as a function of time x(t), we can find the velocity v(t_1)at any time t_1 by calculating the slope of the tangent line to the graph at t=t_1. Therefore if we have a particle’s position described by a function of time, it will be fruitful to derive a method for finding the slope of a tangent line at an arbitrary point.

This idea, called the derivative, is the central concept of differential calculus. Let’s begin our exploration of the derivative with the function f(x)=x^2, the graph of which is shown on the right, and the goal of finding the slope of a tangent to the point A=(2,4), shown on the graph.

 Graph of y=f(x)=x^2 with tangent line to (2,4)
The graph of y=f(x)=x^2 with the tangent line to point A=(2,4) drawn.

Two points determine a line, of course. The problem with finding slopes of tangents to curves is that the slope of a line through any point seemingly must also depend on the other point chosen to determine the line. Therefore, we might begin by choosing an additional point B=(3,9). Then we can find the slope mAB by noting that the slope between any two points is m=delta y/delta x. Therefore mAB=(9-4)/(3-2)=5. However, this is clearly not the slope of a tangent to point A since point B is quite far away from A.

 Graph of y=f(x)=x^2 with secant line AB
The graph of y=f(x)=x^2
with the secant line AB.

We thus select another point C=(2.5,6.25) that is closer to A. We recalculate our slope mAC=(6.25-4)/(2.5-2)=4.5, but again, this is not the slope of a tangent to A.

 Graph of y=f(x)=x^2 with secant line AC
The graph of y=f(x)=x^2
with the secant line AC.

Getting closer, we select D=(2.25,5.0625) and find mAD=(5.0625-4)/(2.25-2)=4.25.

 Graph of y=f(x)=x^2 with secant line AD
The graph of y=f(x)=x^2
with the secant line AD.

Continuing in a similar fashion, mAE=(4.515625-4)/(2.125-2)=4.125 and mAF=(4.25390625-4)/(2.0625-2)=4.0625. As we proceed through the alphabet for our second point, the slope of the line from A to that point gets closer and closer to 4. We might then (correctly, as we will soon show) infer that the slope of the tangent line to A=(2,4) on the graph of y=f(x)=x^2 is 4.

 Graph of y=f(x)=x^2 with secant line AE
The graph of y=f(x)=x^2
with the secant line AE.
Graph of y=f(x)=x^2 with secant line AF
The graph of y=f(x)=x^2
with the secant line AF.

Let’s try to establish a more general formula for the slope of the tangent to a point P=(x,x^2) on the same graph. We consider another arbitrary point Q=(a,a^2) in our calculations. The slope of PQ is mPQ=(x^2-a^2)/(x-a). We can determine an exact value for the slope of the tangent to P by making x and a get nearer and nearer to each other; since x must be fixed, we make x approach a—a limit!

So the slope of the tangent to P is mP=lim(x->a)(x^2-a^2)/(x-a). Now, we set about evaluating that limit. We can factor the numerator as a difference of squares: x^2-a^2=(x+a)(x-a). This leaves lim(x->a)((x-a)(x+a))/(x-a), so a factor of x-a cancels to give lim(x->a)(x+a). At this point, we can simply plug in x for a.

Now we have lim(x->a)(x+a)=lim(x->a)(x+x)=2x. So the slope of a tangent to y=f(x)=x^2 at any point (x,x^2) is 2x.

Some notation and terminology is now in order. We’ve just shown that the derivative of f(x)=x^2 is 2x. We can express the derivative of a function f(x) in several ways:

  • d/dx f or df/dx, where the denominator indicates that the derivative is taken with respect to x—this is known as Leibniz’s notation. We then notate the slope of a tangent to the point where x=a as df/dx|x=a.
  • f’(x) or just f’, where in the latter it is implied that the derivative is taken with respect to x —this is known as Lagrange’s notation. We then notate the slope of a tangent to the point where x=a as f’(a).
  • D_x f or just Df, a notation much more common in differential calculus of several variables and the study of differential equations—this is known as Euler’s notation. We then notate the slope of a tangent to the point where x=a as D_x f(a).
  • f with a dot over it, a notation typically seen only in the study of mechanics and only for derivatives with respect to time—this is known as Newton’s notation.

While the most intuitive way to compute a derivative, taking into account geometric considerations, is to express it as df/dx=lim(x->a)(f(x)-f(a))/(x-a) as we just did with f(x)=x^2, an equivalent though slightly less intuitive expression is often used. If we let h=x-a, then x=a+h, and we can express (f(x)-f(a))/(x-a) as (f(a+h)-f(a))/h. Now x approaches a is equivalent to h approaches 0 since lim(x->a)(x-a)=0 and x-a=h, so df/dx=lim(h->0)(f(a+h)-f(a))/h.

In the next lesson, we will discuss methods for finding derivatives of various functions. In the meantime, chew on this: the derivative is a new kind of operator. Previous operators you have encountered, such as multiplication and addition and even exponentiation, output a number, but the derivative takes in a function and outputs a function.

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