2.1. ELEMENTARY DERIVATIVE RULES
93
sight of the fact that all of the meaning of the derivative still holds that we developed
in Chapter 1. That is, anytime we compute a derivative, that derivative measures the
instantaneous rate of change of the original function, as well as the slope of the tangent
line at any selected point on the curve. The following activity asks you to combine the
just-developed derivative rules with some key perspectives that we studied in Chapter 1.
Activity 2.3.
Each of the following questions asks you to use derivatives to answer key questions
about functions. Be sure to think carefully about each question and to use proper
notation in your responses.
(a) Find the slope of the tangent line to h(z) =
√
z +
1
z at the point where z = 4.
(b) A population of cells is growing in such a way that its total number in millions
is given by the function P(t) = 2(1.37) t + 32, where t is measured in days.
i. Determine the instantaneous rate at which the population is growing on
day 4, and include units on your answer.
ii. Is the population growing at an increasing rate or growing at a decreasing
rate on day 4? Explain.
(c) Find an equation for the tangent line to the curve p(a) = 3a 4 −2a 3 +7a 2 − a +12
at the point where a = −1.
(d) What is the difference between being asked to find the slope of the tangent line
(asked in (a)) and the equation of the tangent line (asked in (c))?
⊳
Summary
In this section, we encountered the following important ideas:
• Given a differentiable function y = f (x), we can express the derivative of f in several
different notations: f ′ (x),
d f
dx ,
dy
dx , and
d
dx [ f (x)].
• The limit definition of the derivative leads to patterns among certain families of
functions that enable us to compute derivative formulas without resorting directly to
the limit definition. For example, if f is a power function of the form f (x) = x n , then
f ′ (x) = nx n−1 for any real number n other than 0. This is called the Rule for Power
Functions.
• We have stated a rule for derivatives of exponential functions in the same spirit as
the rule for power functions: for any positive real number a, if f (x) = a x , then
f ′ (x) = a x ln(a).
• If we are given a constant multiple of a function whose derivative we know, or a sum of
functions whose derivatives we know, the Constant Multiple and Sum Rules make it
93
sight of the fact that all of the meaning of the derivative still holds that we developed
in Chapter 1. That is, anytime we compute a derivative, that derivative measures the
instantaneous rate of change of the original function, as well as the slope of the tangent
line at any selected point on the curve. The following activity asks you to combine the
just-developed derivative rules with some key perspectives that we studied in Chapter 1.
Activity 2.3.
Each of the following questions asks you to use derivatives to answer key questions
about functions. Be sure to think carefully about each question and to use proper
notation in your responses.
(a) Find the slope of the tangent line to h(z) =
√
z +
1
z at the point where z = 4.
(b) A population of cells is growing in such a way that its total number in millions
is given by the function P(t) = 2(1.37) t + 32, where t is measured in days.
i. Determine the instantaneous rate at which the population is growing on
day 4, and include units on your answer.
ii. Is the population growing at an increasing rate or growing at a decreasing
rate on day 4? Explain.
(c) Find an equation for the tangent line to the curve p(a) = 3a 4 −2a 3 +7a 2 − a +12
at the point where a = −1.
(d) What is the difference between being asked to find the slope of the tangent line
(asked in (a)) and the equation of the tangent line (asked in (c))?
⊳
Summary
In this section, we encountered the following important ideas:
• Given a differentiable function y = f (x), we can express the derivative of f in several
different notations: f ′ (x),
d f
dx ,
dy
dx , and
d
dx [ f (x)].
• The limit definition of the derivative leads to patterns among certain families of
functions that enable us to compute derivative formulas without resorting directly to
the limit definition. For example, if f is a power function of the form f (x) = x n , then
f ′ (x) = nx n−1 for any real number n other than 0. This is called the Rule for Power
Functions.
• We have stated a rule for derivatives of exponential functions in the same spirit as
the rule for power functions: for any positive real number a, if f (x) = a x , then
f ′ (x) = a x ln(a).
• If we are given a constant multiple of a function whose derivative we know, or a sum of
functions whose derivatives we know, the Constant Multiple and Sum Rules make it