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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 to compute a formula for
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What is the derivative of a power function of the form What is the derivative of an exponential function of form
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If we know the derivative of what is the derivative of where is a constant?
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If we know the derivatives of and how do we compute the derivative of
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 of a function measures the instantaneous rate of change of with respect to The derivative also tells us the slope of the tangent line to at any given value of 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,
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 quickly, without using the limit definition directly. For example, we would like to apply shortcuts to differentiate a function such as
Preview Activity 2.1.1.
Functions of the form where 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
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Use the limit definition of the derivative to find for
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Use the limit definition of the derivative to find for
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Use the limit definition of the derivative to find for (Hint: Apply this rule to 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 Of
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Conjecture a formula for the derivative of that holds for any positive integer That is, given where is a positive integer, what do you think is the formula for
Subsection 2.1.1 Some Key Notation
In addition to our usual 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 and we sometimes denote the derivative of with respect to by the symbol
which we read “dee-y dee-x.” For example, if we’ll write that the derivative is This notation comes from the fact that the derivative is related to the slope of a line, and slope is measured by Note that while we read as “change in over change in ” we view 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,
means “take the derivative of the quantity in with respect to ” For example, we may write
It is important to note that the independent variable can be different from If we have we then write Similarly, if we say And it is also true that This notation may also be used for second derivatives:
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 is a constant function, its graph is a horizontal line with slope zero at every point. Thus, We summarize this with the following rule.
Constant Functions.
For any real number if then
Example 2.1.1.
If then Similarly,
In your work in Preview Activity 2.1.1, you conjectured that for any positive integer if then This rule can be formally proved for any positive integer and also for any nonzero real number (positive or negative).
Power Functions.
For any nonzero real number if then
Example 2.1.2.
Using the rule for power functions, we can compute the following derivatives. If then Similarly, if then and
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 if then
Example 2.1.3.
If then Similarly, for It is especially important to note that when where is the base of the natural logarithm function, we have that
since This is an extremely important property of the function 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 while in exponential functions, the variable is in the power, as in 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 you should write “” or “” as part of your response.
10.
Let and be differentiable functions for which the following information is known:
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Let be the new function defined by the rule Determine and
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Find an equation for the tangent line to at the point
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Let be the function defined by the rule Is increasing, decreasing, or neither at Why?
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Estimate the value of by using the local linearization of at the point
11.
Let functions and 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 is not differentiable? At what values of is not differentiable? Why?
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Let At what values of is not differentiable? Why?
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Determine and
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Find an equation for the tangent line to at the point