2.6. DERIVATIVES OF INVERSE FUNCTIONS
129
2.6 Derivatives of Inverse Functions
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is the derivative of the natural logarithm function?
• What are the derivatives of the inverse trigonometric functions arcsin(x) and
arctan(x)?
• If g is the inverse of a differentiable function f , how is g ′ computed in terms of f ,
f ′ , and g?
Introduction
Much of mathematics centers on the notion of function. Indeed, throughout our study
of calculus, we are investigating the behavior of functions, often doing so 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? Connected to this question, we
now also ask: if we know how fast a particular process is changing, can we determine how
fast the inverse process is changing?
As we have noted, one of the most important functions in all of mathematics is the
natural exponential function f (x) = e x . Because the natural logarithm, g(x) = ln(x), is the
inverse of the natural exponential function, the natural logarithm is similarly important.
One of our goals in this section is to learn how to differentiate the logarithm function,
and thus expand our library of basic functions with known derivative formulas. First,
we investigate a more familiar setting to refresh some of the basic concepts surrounding
functions and their inverses.
Preview Activity 2.6. The equation y =
5
9 (x − 32) relates a temperature given in x
degrees Fahrenheit to the corresponding temperature y measured in degrees Celcius.
(a) Solve the equation y =
5
9 (x − 32) for x to write x (Fahrenheit temperature) in terms
of y (Celcius temperature).
(b) Let C(x) =
5
9 (x − 32) be the function that takes a Fahrenheit temperature as input
and produces the Celcius temperature as output. In addition, let F(y) be the
function that converts a temperature given in y degrees Celcius to the temperature
F(y) measured in degrees Fahrenheit. Use your work in (a) to write a formula for
F(y).
129
2.6 Derivatives of Inverse Functions
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is the derivative of the natural logarithm function?
• What are the derivatives of the inverse trigonometric functions arcsin(x) and
arctan(x)?
• If g is the inverse of a differentiable function f , how is g ′ computed in terms of f ,
f ′ , and g?
Introduction
Much of mathematics centers on the notion of function. Indeed, throughout our study
of calculus, we are investigating the behavior of functions, often doing so 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? Connected to this question, we
now also ask: if we know how fast a particular process is changing, can we determine how
fast the inverse process is changing?
As we have noted, one of the most important functions in all of mathematics is the
natural exponential function f (x) = e x . Because the natural logarithm, g(x) = ln(x), is the
inverse of the natural exponential function, the natural logarithm is similarly important.
One of our goals in this section is to learn how to differentiate the logarithm function,
and thus expand our library of basic functions with known derivative formulas. First,
we investigate a more familiar setting to refresh some of the basic concepts surrounding
functions and their inverses.
Preview Activity 2.6. The equation y =
5
9 (x − 32) relates a temperature given in x
degrees Fahrenheit to the corresponding temperature y measured in degrees Celcius.
(a) Solve the equation y =
5
9 (x − 32) for x to write x (Fahrenheit temperature) in terms
of y (Celcius temperature).
(b) Let C(x) =
5
9 (x − 32) be the function that takes a Fahrenheit temperature as input
and produces the Celcius temperature as output. In addition, let F(y) be the
function that converts a temperature given in y degrees Celcius to the temperature
F(y) measured in degrees Fahrenheit. Use your work in (a) to write a formula for
F(y).