90
2.1. ELEMENTARY DERIVATIVE RULES
As we next turn to thinking about derivatives of combinations of basic functions, it will
be instructive to have one more type of basic function whose derivative formula we know.
For now, we simply state this rule without explanation or justification; we will explore why
this rule is true in one of the exercises at the end of this section, plus we will encounter
graphical reasoning for why the rule is plausible in Preview Activity 2.2.
Exponential Functions: For any positive real number a, if f (x) = a x , then f ′ (x) =
a x ln(a).
For instance, this rule tells us that if f (x) = 2 x , then f ′ (x) = 2 x ln(2). Similarly, for
p(t) = 10 t , p ′ (t) = 10 t ln(10). It is especially important to note that when a = e, where e
is the base of the natural logarithm function, we have that
d
dx
[e
x ] = e
x ln(e) = e
x
since ln(e) = 1. This is an extremely important property of the function e x : its derivative
function is itself!
Finally, note carefully the distinction between power functions and exponential functions: in power functions, the variable is in the base, as in x 2 , while in exponential
functions, the variable is in the power, as in 2 x . As we can see from the rules, this makes
a big difference in the form of the derivative.
The following activity will check your understanding of the derivatives of the three
basic types of functions noted above.
Activity 2.1.
Use the three rules above to determine the derivative of each of the following functions.
For each, state your answer using full and proper notation, labeling the derivative with
its name. For example, if you are given a function h(z), you should write “h ′ (z) =” or
“
dh
dz =” as part of your response.
(a) f (t) = π
(b) g(z) = 7 z
(c) h(w) = w 3/4
(d) p(x) = 3 1/2
(e) r(t) = (
√
2) t
(f)
d
dq [q −1 ]
(g) m(t) =
1
t 3
⊳
2.1. ELEMENTARY DERIVATIVE RULES
As we next turn to thinking about derivatives of combinations of basic functions, it will
be instructive to have one more type of basic function whose derivative formula we know.
For now, we simply state this rule without explanation or justification; we will explore why
this rule is true in one of the exercises at the end of this section, plus we will encounter
graphical reasoning for why the rule is plausible in Preview Activity 2.2.
Exponential Functions: For any positive real number a, if f (x) = a x , then f ′ (x) =
a x ln(a).
For instance, this rule tells us that if f (x) = 2 x , then f ′ (x) = 2 x ln(2). Similarly, for
p(t) = 10 t , p ′ (t) = 10 t ln(10). It is especially important to note that when a = e, where e
is the base of the natural logarithm function, we have that
d
dx
[e
x ] = e
x ln(e) = e
x
since ln(e) = 1. This is an extremely important property of the function e x : its derivative
function is itself!
Finally, note carefully the distinction between power functions and exponential functions: in power functions, the variable is in the base, as in x 2 , while in exponential
functions, the variable is in the power, as in 2 x . As we can see from the rules, this makes
a big difference in the form of the derivative.
The following activity will check your understanding of the derivatives of the three
basic types of functions noted above.
Activity 2.1.
Use the three rules above to determine the derivative of each of the following functions.
For each, state your answer using full and proper notation, labeling the derivative with
its name. For example, if you are given a function h(z), you should write “h ′ (z) =” or
“
dh
dz =” as part of your response.
(a) f (t) = π
(b) g(z) = 7 z
(c) h(w) = w 3/4
(d) p(x) = 3 1/2
(e) r(t) = (
√
2) t
(f)
d
dq [q −1 ]
(g) m(t) =
1
t 3
⊳