92
2.1. ELEMENTARY DERIVATIVE RULES
Rules together tell us that
d
dx [ f (x) + (−1 · g(x))] = f ′ (x) − g ′ (x), or that “the derivative of
a difference is the difference of the derivatives.” Hence we can now compute derivatives of
sums and differences of elementary functions. For instance,
d
dw (2 w + w 2 ) = 2 w ln(2) + 2w,
and if h(q) = 3q 6 − 4q −3 , then h ′ (q) = 3(6q 5 ) − 4(−3q −4 ) = 18q 5 + 12q −4 .
Activity 2.2.
Use only the rules for constant, power, and exponential functions, together with the
Constant Multiple and Sum Rules, to compute the derivative of each function below
with respect to the given independent variable. Note well that we do not yet know any
rules for how to differentiate the product or quotient of functions. This means that
you may have to do some algebra first on the functions below before you can actually
use existing rules to compute the desired derivative formula. In each case, label the
derivative you calculate with its name using proper notation such as f ′ (x), h ′ (z), dr/dt,
etc.
(a) f (x) = x 5/3 − x 4 + 2 x
(b) g(x) = 14e x + 3x 5 − x
(c) h(z) =
√
z +
1
z 4 + 5 z
(d) r(t) =
√
53 t 7 − πe t + e 4
(e) s(y) = (y 2 + 1)(y 2 − 1)
(f) q(x) =
x 3 − x + 2
x
(g) p(a) = 3a 4 − 2a 3 + 7a 2 − a + 12
⊳
In the same way that we have shortcut rules to help us find derivatives, we introduce
some language that is simpler and shorter. Often, rather than say “take the derivative of
f ,” we’ll instead say simply “differentiate f .” This phrasing is tied to the notion of having
a derivative to begin with: if the derivative exists at a point, we say “ f is differentiable,”
which is tied to the fact that f can be differentiated.
As we work more and more with the algebraic structure of functions, it is important to
strive to develop a big picture view of what we are doing. Here, we can note several general
observations based on the rules we have so far. One is that the derivative of any polynomial
function will be another polynomial function, and that the degree of the derivative is one
less than the degree of the original function. For instance, if p(t) = 7t 5 − 4t 3 + 8t, p is a
degree 5 polynomial, and its derivative, p ′ (t) = 35t 4 − 12t 2 + 8, is a degree 4 polynomial.
Additionally, the derivative of any exponential function is another exponential function:
for example, if g(z) = 7 · 2 z , then g ′ (z) = 7 · 2 z ln(2), which is also exponential.
Furthermore, while our current emphasis is on learning shortcut rules for finding
derivatives without directly using the limit definition, we should be certain not to lose
2.1. ELEMENTARY DERIVATIVE RULES
Rules together tell us that
d
dx [ f (x) + (−1 · g(x))] = f ′ (x) − g ′ (x), or that “the derivative of
a difference is the difference of the derivatives.” Hence we can now compute derivatives of
sums and differences of elementary functions. For instance,
d
dw (2 w + w 2 ) = 2 w ln(2) + 2w,
and if h(q) = 3q 6 − 4q −3 , then h ′ (q) = 3(6q 5 ) − 4(−3q −4 ) = 18q 5 + 12q −4 .
Activity 2.2.
Use only the rules for constant, power, and exponential functions, together with the
Constant Multiple and Sum Rules, to compute the derivative of each function below
with respect to the given independent variable. Note well that we do not yet know any
rules for how to differentiate the product or quotient of functions. This means that
you may have to do some algebra first on the functions below before you can actually
use existing rules to compute the desired derivative formula. In each case, label the
derivative you calculate with its name using proper notation such as f ′ (x), h ′ (z), dr/dt,
etc.
(a) f (x) = x 5/3 − x 4 + 2 x
(b) g(x) = 14e x + 3x 5 − x
(c) h(z) =
√
z +
1
z 4 + 5 z
(d) r(t) =
√
53 t 7 − πe t + e 4
(e) s(y) = (y 2 + 1)(y 2 − 1)
(f) q(x) =
x 3 − x + 2
x
(g) p(a) = 3a 4 − 2a 3 + 7a 2 − a + 12
⊳
In the same way that we have shortcut rules to help us find derivatives, we introduce
some language that is simpler and shorter. Often, rather than say “take the derivative of
f ,” we’ll instead say simply “differentiate f .” This phrasing is tied to the notion of having
a derivative to begin with: if the derivative exists at a point, we say “ f is differentiable,”
which is tied to the fact that f can be differentiated.
As we work more and more with the algebraic structure of functions, it is important to
strive to develop a big picture view of what we are doing. Here, we can note several general
observations based on the rules we have so far. One is that the derivative of any polynomial
function will be another polynomial function, and that the degree of the derivative is one
less than the degree of the original function. For instance, if p(t) = 7t 5 − 4t 3 + 8t, p is a
degree 5 polynomial, and its derivative, p ′ (t) = 35t 4 − 12t 2 + 8, is a degree 4 polynomial.
Additionally, the derivative of any exponential function is another exponential function:
for example, if g(z) = 7 · 2 z , then g ′ (z) = 7 · 2 z ln(2), which is also exponential.
Furthermore, while our current emphasis is on learning shortcut rules for finding
derivatives without directly using the limit definition, we should be certain not to lose