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3 DERIVATIVES

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DERIVATIVES If it were always necessary to compute derivatives directly from the definition, as we did in the Section 3.2, then Such computations would be tedious. The evaluation of some limits would require ingenuity.

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DERIVATIVES Fortunately, several rules have been developed for finding derivatives without having to use the definition directly. These formulas greatly simplify the task of differentiation.

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3.3 Differentiation Formulas

DERIVATIVES 3.3 Differentiation Formulas In this section, we will learn: How to differentiate constant functions, power functions, polynomials, and exponential functions.

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CONSTANT FUNCTION Let’s start with the simplest of all functions—the constant function f(x) = c.

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CONSTANT FUNCTION The graph of this function is the horizontal line y = c, which has slope 0. So, we must have f’(x) = 0.

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A formal proof—from the definition of a derivative—is also easy.

CONSTANT FUNCTION A formal proof—from the definition of a derivative—is also easy.

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In Leibniz notation, we write this rule as follows.

CONSTANT FUNCTION—DERIVATIVE In Leibniz notation, we write this rule as follows.

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POWER FUNCTIONS We next look at the functions f(x) = xn, where n is a positive integer.

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If n = 1, the graph of f(x) = x is the line y = x, which has slope 1.

POWER FUNCTIONS Equation 1 If n = 1, the graph of f(x) = x is the line y = x, which has slope 1. So, You can also verify Equation 1 from the definition of a derivative.

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We have already investigated the cases n = 2 and n = 3.

POWER FUNCTIONS Equation 2 We have already investigated the cases n = 2 and n = 3. In fact, in Section 3.2, we found that:

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For n = 4, we find the derivative of f(x) = x4 as follows:

POWER FUNCTIONS For n = 4, we find the derivative of f(x) = x4 as follows:

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POWER FUNCTIONS Equation 3 Thus,

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Comparing Equations 1, 2, and 3, we see a pattern emerging.

POWER FUNCTIONS Comparing Equations 1, 2, and 3, we see a pattern emerging. It seems to be a reasonable guess that, when n is a positive integer, (d/dx)(xn) = nxn – 1. This turns out to be true. We prove it in two ways; the second proof uses the Binomial Theorem.

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If n is a positive integer, then

POWER RULE If n is a positive integer, then

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POWER RULE Proof 1 The formula can be verified simply by multiplying out the right-hand side (or by summing the second factor as a geometric series).

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POWER RULE Proof 1 If f(x) = xn, we can use Equation 5 in Section 3.1 for f’(a) and the previous equation to write:

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In finding the derivative of x4, we had to expand (x + h)4.

POWER RULE Proof 2 In finding the derivative of x4, we had to expand (x + h)4. Here, we need to expand (x + h)n . To do so, we use the Binomial Theorem—as follows.

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POWER RULE Proof 2 This is because every term except the first has h as a factor and therefore approaches 0.

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We illustrate the Power Rule using various notations in Example 1.

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If f(x) = x6, then f’(x) = 6×5 If y = x1000, then y’ = 1000×999

POWER RULE Example 1 If f(x) = x6, then f’(x) = 6×5 If y = x1000, then y’ = 1000×999 If y = t4, then = 3r2

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NEW DERIVATIVES FROM OLD

When new functions are formed from old functions by addition, subtraction, or multiplication by a constant, their derivatives can be calculated in terms of derivatives of the old functions. In particular, the following formula says that the derivative of a constant times a function is the constant times the derivative of the function.

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If c is a constant and f is a differentiable function, then

CONSTANT MULTIPLE RULE If c is a constant and f is a differentiable function, then

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CONSTANT MULTIPLE RULE

Proof Let g(x) = cf(x). Then,

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NEW DERIVATIVES FROM OLD

Example 2

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NEW DERIVATIVES FROM OLD

The next rule tells us that the derivative of a sum of functions is the sum of the derivatives.

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If f and g are both differentiable, then

SUM RULE If f and g are both differentiable, then

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Let F(x) = f(x) + g(x). Then,

SUM RULE Proof Let F(x) = f(x) + g(x). Then,

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The Sum Rule can be extended to the sum of any number of functions.

For instance, using this theorem twice, we get:

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NEW DERIVATIVES FROM OLD

By writing f – g as f + (-1)g and applying the Sum Rule and the Constant Multiple Rule, we get the following formula.

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If f and g are both differentiable, then

DIFFERENCE RULE If f and g are both differentiable, then

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NEW DERIVATIVES FROM OLD

The Constant Multiple Rule, the Sum Rule, and the Difference Rule can be combined with the Power Rule to differentiate any polynomial—as the following examples demonstrate.

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NEW DERIVATIVES FROM OLD

Example 3

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NEW DERIVATIVES FROM OLD

Example 4 Find the points on the curve y = x4 – 6x where the tangent line is horizontal.

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Horizontal tangents occur where the derivative is zero.

NEW DERIVATIVES FROM OLD Example 4 Horizontal tangents occur where the derivative is zero. We have: Thus, dy/dx = 0 if x = 0 or x2 – 3 = 0, that is, x = ± .

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So, the given curve has horizontal tangents when x = 0, , and – .

NEW DERIVATIVES FROM OLD Example 4 So, the given curve has horizontal tangents when x = 0, , and The corresponding points are (0, 4), ( , -5), and (- , -5).

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NEW DERIVATIVES FROM OLD

Example 5 The equation of motion of a particle is s = 2t3 – 5t2 + 3t + 4, where s is measured in centimeters and t in seconds. Find the acceleration as a function of time. What is the acceleration after 2 seconds?

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The velocity and acceleration are:

NEW DERIVATIVES FROM OLD Example 5 The velocity and acceleration are: The acceleration after 2s is: a(2) = 14 cm/s2

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By analogy with the Sum and Difference

NEW DERIVATIVES FROM OLD By analogy with the Sum and Difference Rules, one might be tempted to guess—as Leibniz did three centuries ago—that the derivative of a product is the product of the derivatives. However, we can see that this guess is wrong by looking at a particular example.

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Let f(x) = x and g(x) = x2. NEW DERIVATIVES FROM OLD

Then, the Power Rule gives f’(x) = 1 and g’(x) = 2x. However, (fg)(x) = x3. So, (fg)’(x) =3×2. Thus, (fg)’ ≠ f’g’.

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THE PRODUCT RULE The correct formula was discovered by Leibniz (soon after his false start) and is called the Product Rule.

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If f and g are both differentiable, then:

THE PRODUCT RULE If f and g are both differentiable, then: In words, the Product Rule says: The derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

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THE PRODUCT RULE Proof Let F(x) = f(x)g(x). Then,

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THE PRODUCT RULE Proof In order to evaluate this limit, we would like to separate the functions f and g as in the proof of the Sum Rule. We can achieve this separation by subtracting and adding the term f(x + h)g(x) in the numerator, as follows.

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THE PRODUCT RULE Proof

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Note that because g(x) is a constant with respect to the variable h.

THE PRODUCT RULE Proof Note that because g(x) is a constant with respect to the variable h. Also, since f is differentiable at x, it is continuous at x by Theorem 4 in Section 3.2, and so See Exercise 55 in Section 2.5

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Find F’(x) if F(x) = (6×3)(7×4).

THE PRODUCT RULE Example 6 Find F’(x) if F(x) = (6×3)(7×4). By the Product Rule, we have:

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THE PRODUCT RULE Notice that we could verify the answer to Example 6 directly by first multiplying the factors: Later, though, we will meet functions, such as y = x2 sinx, for which the Product Rule is the only possible method.

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THE PRODUCT RULE Example 7 If h(x) = xg(x) and it is known that g(3) = 5 and g’(3) = 2, find h’(3). Applying the Product Rule, we get: Therefore,

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If f and g are differentiable, then:

THE QUOTIENT RULE If f and g are differentiable, then: In words, the Quotient Rule says: The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

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THE QUOTIENT RULE Proof Let F(x) = f(x)/g(x). Then,

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THE QUOTIENT RULE Proof We can separate f and g in that expression by subtracting and adding the term f(x)g(x) in the numerator:

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Again, g is continuous by Theorem 4 in Section 3.2. Hence,

THE QUOTIENT RULE Proof Again, g is continuous by Theorem 4 in Section 3.2. Hence,

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The theorems of this section show that:

THE QUOTIENT RULE The theorems of this section show that: Any polynomial is differentiable on . Any rational function is differentiable on its domain.

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THE QUOTIENT RULE Furthermore, the Quotient Rule and the other differentiation formulas enable us to compute the derivative of any rational function—as the next example illustrates.

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THE QUOTIENT RULE Example 8 Let

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THE QUOTIENT RULE Example 8 Then,

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THE QUOTIENT RULE We can use a graphing device to check that the answer to Example 8 is plausible. The figure shows the graphs of the function of Example 8 and its derivative. Notice that, when y grows rapidly (near -2), y’ is large. When y grows slowly, y’ is near 0.

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Don’t use the Quotient Rule every time you see a quotient.

NOTE Don’t use the Quotient Rule every time you see a quotient. Sometimes, it’s easier to rewrite a quotient first to put it in a form that is simpler for the purpose of differentiation.

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NOTE For instance: It is possible to differentiate the function using the Quotient Rule. However, it is much easier to perform the division first and write the function as before differentiating.

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GENERAL POWER FUNCTIONS

The Quotient Rule can be used to extend the Power Rule to the case where the exponent is a negative integer.

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If n is a positive integer, then

GENERAL POWER FUNCTIONS If n is a positive integer, then

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GENERAL POWER FUNCTIONS

Proof

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GENERAL POWER FUNCTIONS

Example 9 If y = 1/x, then

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POWER RULE So far, we know that the Power Rule holds if the exponent n is a positive or negative integer. If n = 0, then x0 = 1, which we know has a derivative of 0. Thus, the Power Rule holds for any integer n.

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What if the exponent is a fraction?

FRACTIONS What if the exponent is a fraction? In Example 3 in Section 3.2, we found that: This can be written as:

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This shows that the Power Rule is true even when n = ½.

FRACTIONS This shows that the Power Rule is true even when n = ½. In fact, it also holds for any real number n, as we will prove in Chapter 7.

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If n is any real number, then

POWER RULE—GENERAL VERSION If n is any real number, then

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a. If f(x) = xπ, then f ’(x) = πxπ-1. b.

POWER RULE Example 10 a. If f(x) = xπ, then f ’(x) = πxπ-1. b.

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Differentiate the function

PRODUCT RULE Example 11 Differentiate the function Here, a and b are constants. It is customary in mathematics to use letters near the beginning of the alphabet to represent constants and letters near the end of the alphabet to represent variables.

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Using the Product Rule, we have:

E. g. 11—Solution 1 Using the Product Rule, we have:

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LAWS OF EXPONENTS E. g. 11—Solution 2 If we first use the laws of exponents to rewrite f(t), then we can proceed directly without using the Product Rule. This is equivalent to the answer in Solution 1.

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TANGENT LINES The differentiation rules enables us to find tangent lines without having to resort to the definition of a derivative.

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They also enables us to find normal lines.

The normal line to a curve C at a point P is the line through P that is perpendicular to the tangent line at P. In the study of optics, one needs to consider the angle between a light ray and the normal line to a lens.

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TANGENT AND NORMAL LINES

Example 12 Find equations of the tangent line and normal line to the curve at the point (1, ½).

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According to the Quotient Rule, we have:

TANGENT LINE Example 12 According to the Quotient Rule, we have:

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So, the slope of the tangent line at (1, ½) is:

Example 12 So, the slope of the tangent line at (1, ½) is: We use the point-slope form to write an equation of the tangent line at (1, ½):

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NORMAL LINE Example 12 The slope of the normal line at (1, ½) is the negative reciprocal of -¼, namely 4. Thus, an equation of the normal line is:

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The curve and its tangent and normal lines are graphed in the figure.

Example 12 The curve and its tangent and normal lines are graphed in the figure. © Thomson Higher Education

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TANGENT LINE Example 13 At what points on the hyperbola xy = 12 is the tangent line parallel to the line 3x + y = 0? Since xy = 12 can be written as y = 12/x, we have:

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Let the x-coordinate of one of the points in question be a.

TANGENT LINE Example 13 Let the x-coordinate of one of the points in question be a. Then, the slope of the tangent line at that point is 12/a2. This tangent line will be parallel to the line 3x + y = 0, or y = -3x, if it has the same slope, that is, -3.

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Equating slopes, we get:

TANGENT LINE Example 13 Equating slopes, we get: Therefore, the required points are: (2, 6) and (-2, -6)

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The hyperbola and the tangents are shown in the figure.

TANGENT LINE Example 13 The hyperbola and the tangents are shown in the figure. © Thomson Higher Education

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Here’s a summary of the differentiation

DIFFERENTIATION FORMULAS Here’s a summary of the differentiation formulas we have learned so far.

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