88
2.1. ELEMENTARY DERIVATIVE RULES
definition of the derivative.
In this present chapter, we will investigate how the limit definition of the derivative,
f
′ (x) = lim
h→0
f (x + h) − f (x)
h
,
leads to interesting patterns and rules that enable us to quickly find a formula for f ′ (x)
based on the formula for f (x) without using the limit definition directly. For example, we
already know that if f (x) = x, then it follows that f ′ (x) = 1. While we could use the limit
definition of the derivative to confirm this, we know it to be true because f (x) is a linear
function with slope 1 at every value of x. One of our goals is to be able to take standard
functions, say ones such as g(x) = 4x 7 − sin(x) + 3e x , and, based on the algebraic form of
the function, be able to apply shortcuts to almost immediately determine the formula for
g ′ (x).
Preview Activity 2.1. Functions of the form f (x) = x n , where n = 1, 2, 3, . . ., are often
called power functions. The first two questions below revisit work we did earlier in Chapter 1,
and the following questions extend those ideas to higher powers of x.
(a) Use the limit definition of the derivative to find f ′ (x) for f (x) = x 2 .
(b) Use the limit definition of the derivative to find f ′ (x) for f (x) = x 3 .
(c) Use the limit definition of the derivative to find f ′ (x) for f (x) = x 4 . (Hint:
(a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 . Apply this rule to (x + h) 4 within the
limit definition.)
(d) Based on your work in (a), (b), and (c), what do you conjecture is the derivative of
f (x) = x 5 ? Of f (x) = x 13 ?
(e) Conjecture a formula for the derivative of f (x) = x n that holds for any positive
integer n. That is, given f (x) = x n where n is a positive integer, what do you think
is the formula for f ′ (x)?
⊲⊳
Some Key Notation
In addition to our usual f ′ notation for the derivative, there are other ways to symbolically
denote the derivative of a function, as well as the instruction to take the derivative. We
know that if we have a function, say f (x) = x 2 , that we can denote its derivative by f ′ (x),
and we write f ′ (x) = 2x. Equivalently, if we are thinking more about the relationship
between y and x, we sometimes denote the derivative of y with respect to x with the
symbol
dy
dx
2.1. ELEMENTARY DERIVATIVE RULES
definition of the derivative.
In this present chapter, we will investigate how the limit definition of the derivative,
f
′ (x) = lim
h→0
f (x + h) − f (x)
h
,
leads to interesting patterns and rules that enable us to quickly find a formula for f ′ (x)
based on the formula for f (x) without using the limit definition directly. For example, we
already know that if f (x) = x, then it follows that f ′ (x) = 1. While we could use the limit
definition of the derivative to confirm this, we know it to be true because f (x) is a linear
function with slope 1 at every value of x. One of our goals is to be able to take standard
functions, say ones such as g(x) = 4x 7 − sin(x) + 3e x , and, based on the algebraic form of
the function, be able to apply shortcuts to almost immediately determine the formula for
g ′ (x).
Preview Activity 2.1. Functions of the form f (x) = x n , where n = 1, 2, 3, . . ., are often
called power functions. The first two questions below revisit work we did earlier in Chapter 1,
and the following questions extend those ideas to higher powers of x.
(a) Use the limit definition of the derivative to find f ′ (x) for f (x) = x 2 .
(b) Use the limit definition of the derivative to find f ′ (x) for f (x) = x 3 .
(c) Use the limit definition of the derivative to find f ′ (x) for f (x) = x 4 . (Hint:
(a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 . Apply this rule to (x + h) 4 within the
limit definition.)
(d) Based on your work in (a), (b), and (c), what do you conjecture is the derivative of
f (x) = x 5 ? Of f (x) = x 13 ?
(e) Conjecture a formula for the derivative of f (x) = x n that holds for any positive
integer n. That is, given f (x) = x n where n is a positive integer, what do you think
is the formula for f ′ (x)?
⊲⊳
Some Key Notation
In addition to our usual f ′ notation for the derivative, there are other ways to symbolically
denote the derivative of a function, as well as the instruction to take the derivative. We
know that if we have a function, say f (x) = x 2 , that we can denote its derivative by f ′ (x),
and we write f ′ (x) = 2x. Equivalently, if we are thinking more about the relationship
between y and x, we sometimes denote the derivative of y with respect to x with the
symbol
dy
dx