Subsection 2.1.1 Some Key Notation
In addition to our usual \(f'\) notation, there are other ways to denote the derivative of a function, as well as the instruction to take the derivative. If we are thinking about the relationship between \(y\) and \(x\text{,}\) we sometimes denote the derivative of \(y\) with respect to \(x\) by the symbol
\begin{equation*}
\frac{dy}{dx}
\end{equation*}
which we read “dee-y dee-x.” For example, if \(y = x^2\text{,}\) we’ll write that the derivative is \(\frac{dy}{dx} = 2x\text{.}\) This notation comes from the fact that the derivative is related to the slope of a line, and slope is measured by \(\frac{\Delta y}{\Delta x}\text{.}\) Note that while we read \(\frac{\Delta y}{\Delta x}\) as “change in \(y\) over change in \(x\text{,}\)” we view \(\frac{dy}{dx}\) as a single symbol, not a quotient of two quantities.
We use a variant of this notation as the instruction to take the derivative. In particular,
\begin{equation*}
\frac{d}{dx}\left[ \Box \right]
\end{equation*}
means “take the derivative of the quantity in \(\Box\) with respect to \(x\text{.}\)” For example, we may write \(\frac{d}{dx}[x^2] = 2x\text{.}\)
It is important to note that the independent variable can be different from \(x\text{.}\) If we have \(f(z) = z^2\text{,}\) we then write \(f'(z) = 2z\text{.}\) Similarly, if \(y = t^2\text{,}\) we say \(\frac{dy}{dt} = 2t\text{.}\) And it is also true that \(\frac{d}{dq}[q^2] = 2q\text{.}\) This notation may also be used for second derivatives: \(f''(z) = \frac{d}{dz}\left[\frac{df}{dz}\right] = \frac{d^2 f}{dz^2}\text{.}\)
In what follows, we’ll build a repertoire of functions for which we can quickly compute the derivative.
Subsection 2.1.2 Constant, Power, and Exponential Functions
So far, we know the derivative formula for two important classes of functions: constant functions and power functions. If \(f(x) = c\) is a constant function, its graph is a horizontal line with slope zero at every point. Thus, \(\frac{d}{dx}[c] = 0\text{.}\) We summarize this with the following rule.
Constant Functions.
For any real number \(c\text{,}\) if \(f(x) = c\text{,}\) then \(f'(x) = 0\text{.}\)
Example 2.1.1.
If \(f(x) = 7\text{,}\) then \(f'(x) = 0\text{.}\) Similarly, \(\frac{d}{dx} [\sqrt{3}] = 0\text{.}\)
In your work in
Preview Activity 2.1.1, you conjectured that for any positive integer
\(n\text{,}\) if
\(f(x) = x^n\text{,}\) then
\(f'(x) = nx^{n-1}\text{.}\) This rule can be formally proved for any positive integer
\(n\text{,}\) and also for any nonzero real number (positive or negative).
Power Functions.
For any nonzero real number \(n\text{,}\) if \(f(x) = x^n\text{,}\) then \(f'(x) = nx^{n-1}\text{.}\)
Example 2.1.2.
Using the rule for power functions, we can compute the following derivatives. If \(g(z) = z^{-3}\text{,}\) then \(g'(z) = -3z^{-4}\text{.}\) Similarly, if \(h(t) = t^{7/5}\text{,}\) then \(\frac{dh}{dt} = \frac{7}{5}t^{2/5}\text{,}\) and \(\frac{d}{dq} [q^{\pi}] = \pi q^{\pi - 1}\text{.}\)
It will be instructive to have a derivative formula for one more type of basic function. For now, we simply state this rule without explanation or justification; we will explore why this rule is true in one of the exercises. And we will encounter graphical reasoning for why the rule is plausible in
Preview Activity 2.2.1.
Exponential Functions.
For any positive real number \(a\text{,}\) if \(f(x) = a^x\text{,}\) then \(f'(x) = a^x \ln(a)\text{.}\)
Example 2.1.3.
If \(f(x) = 2^x\text{,}\) then \(f'(x) = 2^x \ln(2)\text{.}\) Similarly, for \(p(t) = 10^t\text{,}\) \(p'(t) = 10^t \ln(10)\text{.}\) It is especially important to note that when \(a = e\text{,}\) where \(e\) is the base of the natural logarithm function, we have that
\begin{equation*}
\frac{d}{dx} [e^x] = e^x \ln(e) = e^x
\end{equation*}
since \(\ln(e) = 1\text{.}\) This is an extremely important property of the function \(e^x\text{:}\) its derivative function is itself!
Note carefully the distinction between power functions and exponential functions: in power functions, the variable is in the base, as in \(x^2\text{,}\) while in exponential functions, the variable is in the power, as in \(2^x\text{.}\) As we can see from the rules, this makes a big difference in the form of the derivative.
Activity 2.1.2.
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 \(h(z)\text{,}\) you should write “\(h'(z) =\)” or “\(\frac{dh}{dz} =\)” as part of your response.
\(\displaystyle f(t) = \pi\)
\(\displaystyle g(z) = 7^z\)
\(\displaystyle h(w) = w^{3/4}\)
\(\displaystyle p(x) = 3^{1/2}\)
\(\displaystyle r(t) = (\sqrt{2})^t\)
\(\displaystyle s(q) = q^{-1}\)
\(\displaystyle m(t) = \frac{1}{t^3}\)
Subsection 2.1.3 Constant Multiples and Sums of Functions
Next we will learn how to compute the derivative of a function constructed as an algebraic combination of basic functions. For instance, we’d like to be able to take the derivative of a polynomial function such as
\begin{equation*}
p(t) = 3t^5 - 7t^4 + t^2 - 9\text{,}
\end{equation*}
which is a sum of constant multiples of powers of \(t\text{.}\) To that end, we develop two new rules: the Constant Multiple Rule and the Sum Rule.
How is the derivative of \(y = kf(x)\) related to the derivative of \(y = f(x)\text{?}\) Recall that when we multiply a function by a constant \(k\text{,}\) we vertically stretch the graph by a factor of \(|k|\) (and reflect the graph across \(y = 0\) if \(k \lt 0\)). This vertical stretch affects the slope of the graph, so the slope of the function \(y = kf(x)\) is \(k\) times as steep as the slope of \(y = f(x)\text{.}\) Thus, when we multiply a function by a factor of \(k\text{,}\) we change the value of its derivative by a factor of \(k\) as well.,
The Constant Multiple Rule.
For any real number \(k\text{,}\) if \(f(x)\) is a differentiable function with derivative \(f'(x)\text{,}\) then \(\frac{d}{dx}[k f(x)] = k f'(x)\text{.}\)
In words, this rule says that “the derivative of a constant times a function is the constant times the derivative of the function.”
Example 2.1.4.
If \(g(t) = 3 \cdot 5^t\text{,}\) we have \(g'(t) = 3 \cdot 5^t \ln(5)\text{.}\) Similarly, \(\frac{d}{dz} [5z^{-2}] = 5 (-2z^{-3})\text{.}\)
Next we examine a sum of two functions. If we have \(y = f(x)\) and \(y = g(x)\text{,}\) we can compute a new function \(y = (f+g)(x)\) by adding the outputs of the two functions: \((f+g)(x) = f(x) + g(x)\text{.}\) Not only is the value of the new function the sum of the values of the two known functions, but the slope of the new function is the sum of the slopes of the known functions. Therefore, we arrive at the following Sum Rule for derivatives:
The Sum Rule.
If \(f(x)\) and \(g(x)\) are differentiable functions with derivatives \(f'(x)\) and \(g'(x)\) respectively, then \(\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)\text{.}\)
In words, the Sum Rule tells us that “the derivative of a sum is the sum of the derivatives.” It also tells us that a sum of two differentiable functions is also differentiable. Furthermore, because we can view the difference function \(y = (f-g)(x) = f(x) - g(x)\) as \(y = f(x) + (-1 \cdot g(x))\text{,}\) the Sum Rule and Constant Multiple Rules together tell us that \(\frac{d}{dx}[f(x) + (-1 \cdot g(x))] = f'(x) - g'(x)\text{,}\) or that “the derivative of a difference is the difference of the derivatives.” We can now compute derivatives of sums and differences of elementary functions.
Example 2.1.5.
Using the sum rule, \(\frac{d}{dw} (2^w + w^2) = 2^w \ln(2) + 2w\text{.}\) Using both the sum and constant multiple rules, if \(h(q) = 3q^6 - 4q^{-3}\text{,}\) then \(h'(q) = 3 (6q^5) - 4(-3q^{-4}) = 18q^5 + 12q^{-4}\text{.}\)
Activity 2.1.3.
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 \(f'(x)\text{,}\) \(h'(z)\text{,}\) \(dr/dt\text{,}\) etc.
\(\displaystyle f(x) = x^{5/3} - x^4 + 2^x\)
\(\displaystyle g(x) = 14e^x + 3x^5 - x\)
\(\displaystyle h(z) = \sqrt{z} + \frac{1}{z^4} + 5^z\)
\(\displaystyle r(t) = \sqrt{53} \, t^7 - \pi e^t + e^4\)
\(\displaystyle s(y) = (y^2 + 1)(y^2 - 1)\)
\(\displaystyle q(x) = \frac{x^3 - x + 2}{x}\)
\(\displaystyle p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12\)
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 \(f\text{,}\)” we’ll instead say simply “differentiate \(f\text{.}\)” Similarly, if the derivative exists at a point, we say “\(f\) is differentiable at that point,” or that \(f\) can be differentiated.
As we work with the algebraic structure of functions, it is important to develop a big picture view of what we are doing. Here, we make several general observations based on the rules we have so far.
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 \(p(t) = 7t^5 - 4t^3 + 8t\text{,}\) \(p\) is a degree 5 polynomial, and its derivative, \(p'(t) = 35t^4 - 12t^2 + 8\text{,}\) is a degree 4 polynomial.
The derivative of any exponential function is another exponential function: for example, if \(g(z) = 7 \cdot 2^z\text{,}\) then \(g'(z) = 7 \cdot 2^z \ln(2)\text{,}\) which is also exponential.
We should not lose sight of the fact that all of the meaning of the derivative that we developed in
Chapter 1 still holds. The 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.
Activity 2.1.4.
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.
Find the slope of the tangent line to \(h(z) = \sqrt{z} + \frac{1}{z}\) at the point where \(z = 4\text{.}\)
-
A population of cells is growing in such a way that its total number in millions is given by the function \(P(t) = 2(1.37)^t + 32\text{,}\) where \(t\) is measured in days.
Determine the instantaneous rate at which the population is growing on day 4, and include units on your answer.
Is the population growing at an increasing rate or growing at a decreasing rate on day 4? Explain.
Find an equation for the tangent line to the curve \(p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12\) at the point where \(a=-1\text{.}\)
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))?
