The derivative of sine in radians is cosine and the derivative of cosine in radians is -sine. These important facts can be seen by magnifying the unit circle. We assume that you know the definition of radian measure of angles and the associated fact that is the (x,y)-point on the unit circle at the angle , measured counterclockwise from the x-axis. (See Chapter 28, Section 5 and Figure CD-5.1.)

In this section, we use the informal version of Definition 5.5 and work from the relationship between the sine and cosine and the length along the unit circle shown on Figure CD-5.1. Consider what happens as we move from a point to a nearby point . We magnify the unit circle, noting on Figure CD-5.2 that the more we magnify, the straighter the magnified portion of the circle appears.

Figure CD-5.1: Sine and Cosine as a Point on the Unit Circle

The figure with small appears to be a triangle at magnification . The length of the hypotenuse of the apparent triangle is because we use radian measure. (Degrees are not the distance along a unit circle.) The radii coming from the larger figure appear to meet it at right angles, so the apparent triangle is similar to the large triangle at the left with hypotenuse 1 and sides , . (You may have to do some geometry to convince yourself of this, since the corresponding edges are at right angles to one another.) The sides of the apparent triangle are the differences in sine and cosine, with cosine decreasing - hence a negative sign.

Figure CD-5.2 is the microscopic view of the circle that gives us the results

Figure CD-5.2: Derivatives of sine and cosine

Consider the apparent similarity, comparing the long sides of the two triangles,

For in radians,

Example CD-5.1 Increments of Sine

The most important meaning of the increment formula for is simply that a small piece of the graph is given by the linear equation with slope . For example, suppose that we want to estimate the sine of 29 degrees. We know sine of 30 degrees and we can take the increment of -1 degree, using the "microscope equation." We must first convert to radian measure because the increment formulas above are valid only in radian measure. We take and ,

The computer's approximation of sine of 29 degrees is 0.48481.

Example CD-5.2 Limits of Sine

The previous example can be cast in limit notation as: Find

The solution is to recognize this as a special case of the limit defining the derivative,

Rather than using the increment approximation based on a greatly magnified circle, we could use the exact addition formulas to obtain increments of trig functions. In the case of the sine,

These are exact formulas for the increments, but we need to obtain the differential approximations

to complete the last step in proving the local linear approximation.

The point we wish to illustrate is this:

The differential

that discards the error term .

We know because magnified circles appear straighter and straighter as the magnification increases. This observation gives us two interesting limits. Since

1. (Computer Exercise) A View in the Microscope Sketch the view you would see in a powerful microscope focused on the graph above the point where . Verify your prediction using the program Micro1D.

2. Derivative of Cosine
   Use a powerful microscope to prove that the differential of cosine is minus sine,
   
   

3. • 1. Write an exact formula for using the addition formula for cosine. Compare the exact formula with the increment approximation obtained from the microscopic view of the circle. What is the exact formula for ?
   • 2. Approximate the value of cosine of 46 degrees using the linear increment approximation, discarding the term. Give your answer in terms of exact constants such as and as well as numerically. Compare your approximation with the computer or your scientific calculator's approximation.
   • 3. Use the increment approximation for cosine to show that
     
     

   • (Computer Exercise) on the Computer Run the program DfctLimit to show graphically that the gap errors of , , and tend to zero AS FUNCTIONS OF x for appropriate compact intervals . (HINT: Where are the functions and their derivatives defined?)

The point of the next problem is that we can compute the increments of the tangent function directly. Later you will be able to differentiate with rules from Chapter 6.

Problem CD-5.1 Derivative of Tangent (Optional)

Find the differential of the tangent function by examining an increment in the figures below. The segment on the line x=1 between two rays from the circle is the increment of the tangent, because SOH-CAH-TOA with adjacent side of length 1 gives as the length of the segment on x=1 between the x-axis and the ray.

• 1. The area of the triangle in Figure CD-5.3 with as its tiny vertical side is , because the "height" is 1 for the "base" of .
  Figure CD-5.3: An increment of tangent and the secant Function
• 2. The length of the ray at out to x=1 is . Why?
• 3. Show the lower estimate
  
  by finding the area of the circular sector at the left in Figure CD-5.4. (Note that the area of a circular sector of radius r and angle is .)
  
  Figure CD-5.4: A lower estimate and an upper estimate
• 4. Similarly, show
  
  (What is the radius of the sector on the right in Figure CD-5.4?)
• 5. The area of both of the sectors in Figure CD-5.4 is , with when . Why? When is for ?
• 6. Prove that
  
  provided .

The important functional identities of exponential functions are as follows:

Theorem CD-5.1 Laws of Exponents

For a positive base a>0 and any real numbers p and q

We want to use these properties to show what we need to estimate in order to differentiate log and exponential functions. (Practice with the rules can be found in Chapter 28, Section 4, if your skills are rusty.)

Example CD-5.3 The Exact Increment of y=a^x

We write an exact formula for the difference in terms of a^x,

Notice that the last formula says

The rate of change of y=a^x for a fixed change beginning at x is proportional to a^x,

where the constant of proportionality depends only on the change , not x.

Example CD-5.4 Constant Rate of Change for Linear Functions

Suppose an unknown function f[x] increases by a constant amount k every time x increases by another constant amount h. What sort of function is f[x]? The statement that a constant change of h in input makes a constant change of k in output is

Linear functions change at a constant rate. Exponential functions change by a constant percentage for a constant change in input.

Example CD-5.5 Constant PERCENTAGE Change for Exponential Functions

Suppose an unknown function f[x] increases by a constant percentage every time x increases by a constant h. For example, suppose f[x] increases by a third, 33.3%, every time x increases by 1/2. The change in f is

We try to find an exponential solution, f[x]=a^x, of this functional equation

Example CD-5.6 Percentage Rate of Change as

When gets smaller and smaller, we would like to show that converges to a constant . In other words, when is small, , we would like to find an expression for the difference quotient of exponentials of the form,

Notice that, if the limit converges, the result says the following:

Moreover, the convergence is uniform for bounded x:

The "natural" base plays an important role in solving the problem because the number is the unique number that makes

A mathematically simpler approach is to take the definition:

The function is officially defined to be the unique solution to

This "definition" is based on two good guesses. First, that the derivative of an exponential is proportional to the quantity. We saw substantial evidence for this above. Second, that some special number e makes the constant of proportionality equal to 1. The Project on Direct Computation of the Derivative of Exponentials shows you more details on this guess and gives you a way to compute .

Technically, the approach relies on convergence of Euler's approximation to differential equations. This is easier than the convergence problems in the direct approach to the exponential above and has many other applications. (We have not proved that Euler's approximation converges, but we have seen it work in several examples: S-I-R, the canary, and so forth. The proof is in the Mathematical Background CD.) Once we have made this the "official" definition, we can use Euler's approximation to obtain the specific approximation

Theorem CD-5.2 Derivatives of Logs and Exponentials The derivative of the natural base exponential function is

or, written in terms of the independent variable,

The derivative of the natural base logarithm is

Once we know the derivative of the natural exponential and rules of differentiation, we can find the differentials of all exponentials. For this reason, the natural log and exponential play a major role in science and mathematics. Just as radian measure makes the calculus of trig functions "natural," the base for logs and exponentials makes their calculus "natural."

1. Fixed Percentage Changes
   
   • 1. Find an exponential function f[x]=a^x that doubles every time x increases by 1. Write this English question as a mathematical equation and solve it using properties of exponents.
   • 2. Find an exponential function f[x]=b^x that triples every time x increases by 1. Write this English question as a mathematical equation and solve it using properties of exponents.
   • 3. Find an exponential function f[x]=c^x that increases by 50% every time x increases by 1/2 x-unit. Write this English question as a mathematical equation and solve it using properties of exponents and the logarithm.
   • 4. Show that the exponential base a from the first part is NOT equal to c from the third part. Should not a function that increases by 50% in 1/2 unit be the same as one that increases by 100% in 1 unit? Why not?
   • 5. Let f[x] be an unknown function, h and k unknown constants. Write the statement, "The change in as x increases by h equals k times f[x]." as a mathematical equation. In other words, your equation should say, "f[x] increases by as x increases by h."

   • A Doubling Exponential
     Suppose algae cells in a warm pond double every 6 hours and at time t=0 (hrs) there is one cell. How many cells are there in 6 hours? How many cells are there in 12 hours? How many in 18 hours?

     How many 6-hour periods are there in t hours? (A formula.)

     Give the number of cells n as a function of t,
     

     
     

     Suppose at time t₁ there are a billion cells. How many are there at time t₁+6?

     What is the formula for the rate of growth of algae cells in a 6-hour period beginning at an unknown time t? (Compare your work to Problem CD-28.2 and the program ExpGth of Chapter 28.)

     The next exercise has you practice using the functional identities for the logarithm. The point of the exercise is that we need only one limit, , and the functional identity.

   • The Derivative of Natural Log
     
     • 1. Given that , write the increment approximation for at x=1 to show that
       
       with when .
     • 2. Use properties of logs to show that
       
       
     • 3. Suppose you are given that
       
       with when . Use these two facts to prove that for all positive x, bounded away from 0,
       
       with when .
     • 4. Use the above to show that
       
       

     • What is wrong with the following computation?
       
       

     • Percent Growth and the Natural Base
       
       • 1. We know that exponentials grow at a constant PERCENTAGE rate for fixed steps. (See the program PercentGth.) Let f[x]=e^rx and substitute this into the expression
         
         
       • 2. Suppose you know that with when . Show that when .

       • (Computer Exercise) on the Computer Run the program DfctLimit to show graphically that the gap errors of and tend to zero AS FUNCTIONS OF x for appropriate compact intervals . (HINT: Where are the functions and their derivatives defined?)

         The next exercise has some practice using these derivatives in the increment approximation. Do not use your calculator until you have written the symbolic expressions. (You do not have to use it at all, but you can check your work if you wish.)

       • Natural Increments
         
         • 1. Use the formula for the derivative of the natural exponential to write the increment approximation for y=e^x,
           
           
         • 2. Use the formula for the derivative of the natural logarithm to estimate .

We saw in the Exercise CD-3.2.1 that a function can be continuous but still not smooth or differentiable. An official definition of continuity is the following

Definition: A real function f[x] is continuous at the real point a if f[a] is defined and

Intuitively, this just means that f[x] is close to f[a] when x is close to a, for every , f[x] is defined and

Theorem CD-5.3 Continuity of f[x] and f'[x] If f[x] is smooth on the interval a<x<b, then both f[x] and f'[x] are continuous at every point c in (a,b).

INTUITIVE PROOF FOR f[x]:

Proof of continuity of f is easy algebraically but is obvious geometrically: A graph that is indistinguishable from linear clearly only moves a small amount in a small x-step. Algebraically, we want to show that if then . Take x=x₁ and and use the approximation where is medium times small = small, so . That is the algebraic proof. Draw the picture on a small scale.

INTUITIVE PROOF FOR f'[x]:

Proof of continuity of f'[x] requires us to view the increment from both ends. First take x=x₁ and and use the approximation

NOTE:

The derivative defined in many calculus books is a weaker pointwise notion than the notion of smoothness we have defined. The weak derivative function need not be continuous. (The same approximation does not apply at both ends with the weak definition.) This is explained in the Mathematical Background Chapter on "Epsilon - Delta" Approximations.

1. • 1. Suppose that f[x] is smooth on an interval around a so that the "microscope" increment equation is valid. Suppose that so that for . Show that ; in other words, show that smooth real functions are continuous at real points.
   • 2. Consider the real function f[x]=1/x, which is undefined at x=0. We could extend the definition by simply assigning f[0]=0. Show that this function is not continuous at x=0 but is continuous at every other real x.
   • 3. Give an intuitive graphical description of the definition of continuity in terms of powerful microscopes and explain why it follows that smooth functions must be continuous.
   • 4. The function is defined for ; there is nothing wrong with f[0]. However, our increment computation for above was not valid at x=0 because a microscopic view of the graph focused at x=0 looks like a vertical ray (or half-line). Explain why this is so, but show that f[x] is still continuous "from the right;" that is, if , then but is very large.