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What are alternate notations for the derivative?
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How can we use the algebraic structure of a function [latex]f(x)[/latex] to compute a formula for [latex]f'(x)\text{?}[/latex]
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What is the derivative of a power function of the form [latex]f(x) = x^n\text{?}[/latex] What is the derivative of an exponential function of form [latex]f(x) = a^x\text{?}[/latex]
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If we know the derivative of [latex]y = f(x)\text{,}[/latex] what is the derivative of [latex]y = k f(x)\text{,}[/latex] where [latex]k[/latex] is a constant?
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If we know the derivatives of [latex]y = f(x)[/latex] and [latex]y = g(x)\text{,}[/latex] how do we compute the derivative of [latex]y = f(x) + g(x)\text{?}[/latex]
Math Test: Section 2.1 Elementary derivative rules
Motivating Questions
In Chapter 1, we developed the concept of the derivative of a function. We now know that the derivative [latex]f'[/latex] of a function [latex]f[/latex] measures the instantaneous rate of change of [latex]f[/latex] with respect to [latex]x\text{.}[/latex] The derivative also tells us the slope of the tangent line to [latex]y=f(x)[/latex] at any given value of [latex]x\text{.}[/latex] So far, we have focused on interpreting the derivative graphically or, in the context of a physical setting, as a meaningful rate of change. To calculate the value of the derivative at a specific point, we have relied on the limit definition of the derivative,
\begin{equation*}
f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\text{.}
\end{equation*}
f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\text{.}
\end{equation*}
In this chapter, we investigate how the limit definition of the derivative leads to interesting patterns and rules that enable us to find a formula for [latex]f'(x)[/latex] quickly, without using the limit definition directly. For example, we would like to apply shortcuts to differentiate a function such as [latex]g(x) = 4x^7 - \sin(x) + 3e^x[/latex]
Preview Activity 2.1.1.
Functions of the form [latex]f(x) = x^n\text{,}[/latex] where [latex]n = 1, 2, 3, \ldots\text{,}[/latex] are often called power functions. The first two questions below revisit work we did earlier in Chapter 1, and the following questions extend those ideas to higher powers of [latex]x\text{.}[/latex]
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Use the limit definition of the derivative to find [latex]f'(x)[/latex] for [latex]f(x) = x^2\text{.}[/latex]
-
Use the limit definition of the derivative to find [latex]f'(x)[/latex] for [latex]f(x) = x^3\text{.}[/latex]
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Use the limit definition of the derivative to find [latex]f'(x)[/latex] for [latex]f(x) = x^4\text{.}[/latex] (Hint: [latex](a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\text{.}[/latex] Apply this rule to [latex](x+h)^4[/latex] within the limit definition.)
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Based on your work in (a), (b), and (c), what do you conjecture is the derivative of [latex]f(x) = x^5\text{?}[/latex] Of [latex]f(x) = x^{13}\text{?}[/latex]
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Conjecture a formula for the derivative of [latex]f(x) = x^n[/latex] that holds for any positive integer [latex]n\text{.}[/latex] That is, given [latex]f(x) = x^n[/latex] where [latex]n[/latex] is a positive integer, what do you think is the formula for [latex]f'(x)\text{?}[/latex]
Subsection 2.1.1 Some Key Notation
In addition to our usual [latex]f'[/latex] 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 [latex]y[/latex] and [latex]x\text{,}[/latex] we sometimes denote the derivative of [latex]y[/latex] with respect to [latex]x[/latex] by the symbol
\begin{equation*}
\frac{dy}{dx}
\end{equation*}
\frac{dy}{dx}
\end{equation*}
which we read “dee-y dee-x.” For example, if [latex]y = x^2\text{,}[/latex] we’ll write that the derivative is [latex]\frac{dy}{dx} = 2x\text{.}[/latex] This notation comes from the fact that the derivative is related to the slope of a line, and slope is measured by [latex]\frac{\Delta y}{\Delta x}\text{.}[/latex] Note that while we read [latex]\frac{\Delta y}{\Delta x}[/latex] as “change in [latex]y[/latex] over change in [latex]x\text{,}[/latex]” we view [latex]\frac{dy}{dx}[/latex] 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*}
\frac{d}{dx}\left[ \Box \right]
\end{equation*}
means “take the derivative of the quantity in [latex]\Box[/latex] with respect to [latex]x\text{.}[/latex]” For example, we may write [latex]\frac{d}{dx}[x^2] = 2x\text{.}[/latex]
It is important to note that the independent variable can be different from [latex]x\text{.}[/latex] If we have [latex]f(z) = z^2\text{,}[/latex] we then write [latex]f'(z) = 2z\text{.}[/latex] Similarly, if [latex]y = t^2\text{,}[/latex] we say [latex]\frac{dy}{dt} = 2t\text{.}[/latex] And it is also true that [latex]\frac{d}{dq}[q^2] = 2q\text{.}[/latex] This notation may also be used for second derivatives: [latex]f''(z) = \frac{d}{dz}\left[\frac{df}{dz}\right] = \frac{d^2 f}{dz^2}\text{.}[/latex]
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 [latex]f(x) = c[/latex] is a constant function, its graph is a horizontal line with slope zero at every point. Thus, [latex]\frac{d}{dx}[c] = 0\text{.}[/latex] We summarize this with the following rule.
Constant Functions.
For any real number [latex]c\text{,}[/latex] if [latex]f(x) = c\text{,}[/latex] then [latex]f'(x) = 0\text{.}[/latex]
Example 2.1.1.
If [latex]f(x) = 7\text{,}[/latex] then [latex]f'(x) = 0\text{.}[/latex] Similarly, [latex]\frac{d}{dx} [\sqrt{3}] = 0\text{.}[/latex]
In your work in Preview Activity 2.1.1, you conjectured that for any positive integer [latex]n\text{,}[/latex] if [latex]f(x) = x^n\text{,}[/latex] then [latex]f'(x) = nx^{n-1}\text{.}[/latex] This rule can be formally proved for any positive integer [latex]n\text{,}[/latex] and also for any nonzero real number (positive or negative).
Power Functions.
For any nonzero real number [latex]n\text{,}[/latex] if [latex]f(x) = x^n\text{,}[/latex] then [latex]f'(x) = nx^{n-1}\text{.}[/latex]
Example 2.1.2.
Using the rule for power functions, we can compute the following derivatives. If [latex]g(z) = z^{-3}\text{,}[/latex] then [latex]g'(z) = -3z^{-4}\text{.}[/latex] Similarly, if [latex]h(t) = t^{7/5}\text{,}[/latex] then [latex]\frac{dh}{dt} = \frac{7}{5}t^{2/5}\text{,}[/latex] and [latex]\frac{d}{dq} [q^{\pi}] = \pi q^{\pi - 1}\text{.}[/latex]
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 [latex]a\text{,}[/latex] if [latex]f(x) = a^x\text{,}[/latex] then [latex]f'(x) = a^x \ln(a)\text{.}[/latex]
Example 2.1.3.
If [latex]f(x) = 2^x\text{,}[/latex] then [latex]f'(x) = 2^x \ln(2)\text{.}[/latex] Similarly, for [latex]p(t) = 10^t\text{,}[/latex] [latex]p'(t) = 10^t \ln(10)\text{.}[/latex] It is especially important to note that when [latex]a = e\text{,}[/latex] where [latex]e[/latex] 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*}
\frac{d}{dx} [e^x] = e^x \ln(e) = e^x
\end{equation*}
since [latex]\ln(e) = 1\text{.}[/latex] This is an extremely important property of the function [latex]e^x\text{:}[/latex] 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 [latex]x^2\text{,}[/latex] while in exponential functions, the variable is in the power, as in [latex]2^x\text{.}[/latex] 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 [latex]h(z)\text{,}[/latex] you should write “[latex]h'(z) =[/latex]” or “[latex]\frac{dh}{dz} =[/latex]” as part of your response.
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[latex]f(t) = \pi[/latex]
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[latex]g(z) = 7^z[/latex]
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[latex]h(w) = w^{3/4}[/latex]
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[latex]p(x) = 3^{1/2}[/latex]
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[latex]r(t) = (\sqrt{2})^t[/latex]
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[latex]s(q) = q^{-1}[/latex]
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[latex]m(t) = \frac{1}{t^3}[/latex]
10.
Let [latex]f[/latex] and [latex]g[/latex] be differentiable functions for which the following information is known: [latex]f(2) = 5\text{,}[/latex] [latex]g(2) = -3\text{,}[/latex] [latex]f'(2) = -1/2\text{,}[/latex] [latex]g'(2) = 2\text{.}[/latex]
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Let [latex]h[/latex] be the new function defined by the rule [latex]h(x) = 3f(x) - 4g(x)\text{.}[/latex] Determine [latex]h(2)[/latex] and [latex]h'(2)\text{.}[/latex]
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Find an equation for the tangent line to [latex]y = h(x)[/latex] at the point [latex](2,h(2))\text{.}[/latex]
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Let [latex]p[/latex] be the function defined by the rule [latex]p(x) = -2f(x) + \frac{1}{2}g(x)\text{.}[/latex] Is [latex]p[/latex] increasing, decreasing, or neither at [latex]a = 2\text{?}[/latex] Why?
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Estimate the value of [latex]p(2.03)[/latex] by using the local linearization of [latex]p[/latex] at the point [latex](2,p(2))\text{.}[/latex]
11.
Let functions [latex]p[/latex] and [latex]q[/latex] be the piecewise linear functions given by their respective graphs in Figure 2.1.6. Use the graphs to answer the following questions.
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At what values of [latex]x[/latex] is [latex]p[/latex] not differentiable? At what values of [latex]x[/latex] is [latex]q[/latex] not differentiable? Why?
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Let [latex]r(x) = p(x) + 2q(x)\text{.}[/latex] At what values of [latex]x[/latex] is [latex]r[/latex] not differentiable? Why?
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Determine [latex]r'(-2)[/latex] and [latex]r'(0)\text{.}[/latex]
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Find an equation for the tangent line to [latex]y = r(x)[/latex] at the point [latex](2,r(2))\text{.}[/latex]
