2.1. ELEMENTARY DERIVATIVE RULES
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which we read “dee-y dee-x.” This notation comes from the fact that the derivative is
related to the slope of a line, and slope is measured by
△y
△x . Note that while we read
△y
△x
as “change in y over change in x,” for the derivative symbol
dy
dx , we view this is a single
symbol, not a quotient of two quantities 1 . For example, if y = x 2 , we’ll write that the
derivative is
dy
dx = 2x.
Furthermore, we use a variant of
dy
dx notation to convey the instruction to take the
derivative of a certain quantity with respect to a given variable. In particular, if we write
d
dx
[]
this means “take the derivative of the quantity in with respect to x.” To continue our
example above with the squaring function, here we may write
d
dx [x 2 ] = 2x.
It is important to note that the independent variable can be different from x. If we
have f (z) = z 2 , we then write f ′ (z) = 2z. Similarly, if y = t 2 , we can say
dy
dt = 2t. And
changing the variable and derivative notation once more, it is also true that
d
dq [q 2 ] = 2q.
This notation may also be applied to second derivatives: f ′′ (z) =
d
dz
d f
dz
=
d 2 f
dz 2 .
In what follows, we’ll be working to widely expand our repertoire of functions for
which we can quickly compute the corresponding derivative formula
Constant, Power, and Exponential Functions
So far, we know the derivative formula for two important classes of functions: constant
functions and power functions. For the first kind, observe that if f (x) = c is a constant
function, then its graph is a horizontal line with slope zero at every point. Thus,
d
dx [c] = 0.
We summarize this with the following rule.
Constant Functions: For any real number c, if f (x) = c, then f ′ (x) = 0.
Thus, if f (x) = 7, then f ′ (x) = 0. Similarly,
d
dx [
√
3] = 0.
For power functions, from your work in Preview Activity 2.1, you have conjectured that
for any positive integer n, if f (x) = x n , then f ′ (x) = nx n−1 . Not only can this rule be
formally proved to hold for any positive integer n, but also for any nonzero real number
(positive or negative).
Power Functions: For any nonzero real number, if f (x) = x n , then f ′ (x) = nx n−1 .
This rule for power functions allows us to find derivatives such as the following:
if g(z) = z −3 , then g ′ (z) = −3z −4 . Similarly, if h(t) = t 7/5 , then
dh
dt =
7
5 t 2/5 ; likewise,
d
dq [q π ] = πq π−1 .
1 That is, we do not say “dee-y over dee-x.”
89
which we read “dee-y dee-x.” This notation comes from the fact that the derivative is
related to the slope of a line, and slope is measured by
△y
△x . Note that while we read
△y
△x
as “change in y over change in x,” for the derivative symbol
dy
dx , we view this is a single
symbol, not a quotient of two quantities 1 . For example, if y = x 2 , we’ll write that the
derivative is
dy
dx = 2x.
Furthermore, we use a variant of
dy
dx notation to convey the instruction to take the
derivative of a certain quantity with respect to a given variable. In particular, if we write
d
dx
[]
this means “take the derivative of the quantity in with respect to x.” To continue our
example above with the squaring function, here we may write
d
dx [x 2 ] = 2x.
It is important to note that the independent variable can be different from x. If we
have f (z) = z 2 , we then write f ′ (z) = 2z. Similarly, if y = t 2 , we can say
dy
dt = 2t. And
changing the variable and derivative notation once more, it is also true that
d
dq [q 2 ] = 2q.
This notation may also be applied to second derivatives: f ′′ (z) =
d
dz
d f
dz
=
d 2 f
dz 2 .
In what follows, we’ll be working to widely expand our repertoire of functions for
which we can quickly compute the corresponding derivative formula
Constant, Power, and Exponential Functions
So far, we know the derivative formula for two important classes of functions: constant
functions and power functions. For the first kind, observe that if f (x) = c is a constant
function, then its graph is a horizontal line with slope zero at every point. Thus,
d
dx [c] = 0.
We summarize this with the following rule.
Constant Functions: For any real number c, if f (x) = c, then f ′ (x) = 0.
Thus, if f (x) = 7, then f ′ (x) = 0. Similarly,
d
dx [
√
3] = 0.
For power functions, from your work in Preview Activity 2.1, you have conjectured that
for any positive integer n, if f (x) = x n , then f ′ (x) = nx n−1 . Not only can this rule be
formally proved to hold for any positive integer n, but also for any nonzero real number
(positive or negative).
Power Functions: For any nonzero real number, if f (x) = x n , then f ′ (x) = nx n−1 .
This rule for power functions allows us to find derivatives such as the following:
if g(z) = z −3 , then g ′ (z) = −3z −4 . Similarly, if h(t) = t 7/5 , then
dh
dt =
7
5 t 2/5 ; likewise,
d
dq [q π ] = πq π−1 .
1 That is, we do not say “dee-y over dee-x.”