Much of mathematics centers on the notion of function. Indeed, throughout our study of calculus, we are investigating the behavior of functions, with particular emphasis on how fast the output of the function changes in response to changes in the input. Because each function represents a process, a natural question to ask is whether or not the particular process can be reversed. That is, if we know the output that results from the function, can we determine the input that led to it? And if we know how fast a particular process is changing, can we determine how fast the inverse process is changing?
One of the most important functions in all of mathematics is the natural exponential function . Its inverse, the natural logarithm, , is similarly important. One of our goals in this section is to learn how to differentiate the logarithm function. First, we review some of the basic concepts surrounding functions and their inverses.
b. Now let be the function that takes a Fahrenheit temperature as input and produces the Celcius temperature as output. In addition, let be the function that converts a temperature given in degrees Celcius to the temperature measured in degrees Fahrenheit. Use your work above to write a formula for .
c. Next consider the new function defined by . Use the formulas for and to determine an expression for and simplify this expression as much as possible. What do you observe?
Subsection2.6.1Basic facts about inverse functions
A function is a rule that associates each element in the set to one and only one element in the set . We call the domain of and the codomain of . If there exists a function such that for every possible choice of in the set and for every in the set , then we say that is the inverse of .
We often use the notation (read “-inverse”) to denote the inverse of . The inverse function undoes the work of . Indeed, if , then
.
Thus, the equations and say the same thing. The only difference between the two equations is one of perspective — one is solved for , while the other is solved for .
A function is one-to-one provided that no two distinct inputs lead to the same output.
and onto 2
A function is onto provided that every possible element of the codomain can be realized as an output of the function for some choice of input from the domain.
The last fact reveals a special relationship between the graphs of and . If a point that lies on the graph of , then it is also true that , which means that the point lies on the graph of . This shows us that the graphs of and are the reflections of each other across the line , because this reflection is precisely the geometric action that swaps the coordinates in an ordered pair. In Figure 2.6.2, we see this illustrated by the function and its inverse, with the points and highlighting the reflection of the curves across .
To close our review of important facts about inverses, we recall that the natural exponential function has an inverse function, namely the natural logarithm, . Thus, writing is interchangeable with , plus for every real number and for every positive real number .
Subsection2.6.2The derivative of the natural logarithm function
In what follows, we find a formula for the derivative of . To do so, we take advantage of the fact that we know the derivative of the natural exponential function, the inverse of . In particular, we know that writing is equivalent to writing . Now we differentiate both sides of this equation and observe that
This rule for the natural logarithm function now joins our list of basic derivative rules. Note that this rule applies only to positive values of , as these are the only values for which is defined.
Also notice that for the first time in our work, differentiating a basic function of a particular type has led to a function of a very different nature: the derivative of the natural logarithm is not another logarithm, nor even an exponential function, but rather a rational one.
In Figure 2.6.3, we are reminded that since the natural exponential function has the property that its derivative is itself, the slope of the tangent to is equal to the height of the curve at that point. For instance, at the point , the slope of the tangent line is , and at , the tangent line’s slope is .
At the corresponding points and on the graph of the natural logarithm function (which come from reflecting and across the line ), we know that the slope of the tangent line is the reciprocal of the -coordinate of the point (since ). Thus, at , we have , and at ,.
In particular, we observe that and . This is not a coincidence, but in fact holds for any curve and its inverse, provided the inverse exists. This is due to the reflection across . It changes the roles of and , thus reversing the rise and run, so the slope of the inverse function at the reflected point is the reciprocal of the slope of the original function.
Subsection2.6.3Inverse trigonometric functions and their derivatives
Trigonometric functions are periodic, so they fail to be one-to-one, and thus do not have inverse functions. However, we can restrict the domain of each trigonometric function so that it is one-to-one on that domain.
For instance, consider the sine function on the domain . Because no output of the sine function is repeated on this interval, the function is one-to-one and thus has an inverse. Thus, the function with and codomain has an inverse function such that
We call the arcsine (or inverse sine) function and write . It is especially important to remember that
and
say the same thing. “The arcsine of ” means “the angle whose sine is .” For example, means that is the angle whose sine is , which is equivalent to writing .
Next, we determine the derivative of the arcsine function. Letting , our goal is to find . Since is the angle whose sine is , it is equivalent to write
Finally, we recall that , so the denominator of is the function , or in other words, “the cosine of the angle whose sine is .” A bit of right triangle trigonometry allows us to simplify this expression considerably.
Let’s say that , so that is the angle whose sine is . We can picture as an angle in a right triangle with hypotenuse and a vertical leg of length , as shown in Figure 2.6.5. The horizontal leg must be , by the Pythagorean Theorem.
Differentiate both sides of the equation you found in (a). Solve the resulting equation for , writing as simply as possible in terms of a trigonometric function evaluated at .
While derivatives for other inverse trigonometric functions can be established similarly, for now we limit ourselves to the arcsine and arctangent functions.
Subsection2.6.4The link between the derivative of a function and the derivative of its inverse
In Figure 2.6.3, we saw an interesting relationship between the slopes of tangent lines to the natural exponential and natural logarithm functions at points reflected across the line . In particular, we observed that at the point on the graph of , the slope of the tangent line is , while at the corresponding point on the graph of , the slope of the tangent line is , which is the reciprocal of .
That the two corresponding tangent lines have reciprocal slopes is not a coincidence. If and are differentiable inverse functions, then if and only if , then for every in the domain of . Differentiating both sides of this equation, we have
Solving for , we have . Here we see that the slope of the tangent line to the inverse function at the point is precisely the reciprocal of the slope of the tangent line to the original function at the point .
To see this more clearly, consider the graph of the function shown in Figure 2.6.6, along with its inverse . Given a point that lies on the graph of , we know that lies on the graph of ; because and . Now, applying the rule that to the value , we have
,
which is precisely what we see in the figure: the slope of the tangent line to at is the reciprocal of the slope of the tangent line to at , since these two lines are reflections of one another across the line .
The rules we derived for ,, and are all just specific examples of this general property of the derivative of an inverse function. For example, with and , it follows that