The Derivative - CALCULUS - MATHEMATICS IN HISTORY - Mathematics for the liberal arts

Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 4. CALCULUS

4.4 The Derivative

Recall from algebra class the definition of a function. A function is a rule that assigns to each number x in some domain exactly one number y in a range. For example, y = x2 describes a function by giving a formula. The domain of the function is the set of values you can “put in” the function, all real numbers in this case (since every number can be multiplied by itself). The range is the set of values that you “get out” of the function, and for the squaring function this is the set [0, ∞). You probably remember many happy hours spent finding the domains and ranges of functions.

In addition to having a domain and range, we know that y = x2 describes a function because for any x we put in we get out exactly one y. If x = 3 goes in, y = 9 comes out. Put in x = 0 and you get out y = 0, etc. There is only one way to square a number.

You may remember that functions have graphs that pass the vertical line test. This is the same thing; at any particular x-value, the function should only have a single y-value (which is where it intersects the vertical line).

Not all formulas describe functions. For example, y = x1/2 is not a function, because if you put in x = 9, the value for y is ambiguous. It could be y = 3 or y = – 3. However, usually when we write y = images we mean the positive square root; and this is a function, because there is only one way to find a positive square root.

In Example 4.3 we saw a process that can unambiguously tell us the slope of a tangent line at any point on a curve. That is, there is a formula that can tell us slopes when given x-values. If you name an x-value, I can tell you the slope. Name a different x-value, and I can tell you the new slope.

Since the slope of a tangent line is a rule we can give unambiguously, we know from our experience in algebra that it is a function, and this function has a name. It is called the derivative. Sometimes people refer to the derivative as the “slope generating function.”

Definition. The derivative of a function f(x) is a new function denoted f′(x) defined by

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The process of finding the derivative of a function is called differentiation. The notation f′(x) is read “f prime of x.”

images EXAMPLE 4.4

Let’s find the derivative of f(x) = x3. According to our definition,

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Using the expansion (x + h)3 = x3 + 3x2h + 3xh2 + h3, we have

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We can use this new function to find the slope of the tangent line at any point on x3. For example, when x = 0, the slope is f′(0) = 3(02) = 0. Likewise, when x = 1, the slope is f′(1) = 3(12) = 3. This is represented visually in Figure 4.9.

Figure 4.9 Graphs of f(x) and the derivative f′(x).

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EXERCISES

4.13 Use the definition to find the derivative of the function f(x) = x2 + 3. Draw a sketch of f(x), or graph it on a calculator, and check that the slopes given by f′(x) look right.

4.14 Use the definition to find the derivative of the function f(x) = x2 + 5. Draw a sketch of f(x), or graph it on a calculator, and check that the slopes given by f′(x) look right.

4.15 Use the definition to find the derivative of the function f(x) = x2 + x. Draw a sketch of f(x), or graph it on a calculator, and check that the slopes given by f′(x) look right.

4.16 Use the definition to find the derivative of the function f(x) = 3x. Draw a sketch of f(x), or graph it on a calculator, and check that the slopes given by f′(x) look right.

4.17 Use the definition to find the derivative of the function f(x) = 3x + 1. Draw a sketch of f(x), or graph it on a calculator, and check that the slopes given by f′(x) look right.

4.18 Use the definition to find the derivative of the function f(x) = –2x + 2. Draw a sketch of f(x), or graph it on a calculator, and check that the slopes given by f′(x) look right.

4.19 Use the definition to find the derivative of the function f(x) = x3 + x. Draw a sketch of f(x), or graph it on a calculator, and check that the slopes given by f′(x) look right.