Mathematics
Derivatives of Sin, Cos and Tan
The derivatives of sin, cos, and tan are fundamental concepts in calculus. The derivative of sin(x) is cos(x), the derivative of cos(x) is -sin(x), and the derivative of tan(x) is sec^2(x). These derivatives are used to find the rate of change of trigonometric functions and are essential in solving various mathematical problems involving trigonometric functions.
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7 Key excerpts on "Derivatives of Sin, Cos and Tan"
eBook - PDF- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
33–1 Derivatives of the Sine and Cosine Functions Derivative of sin u Approximated Graphically Before deriving a formula for the derivative of the sine function, let us use a sketch to get an indication of its shape. ◆◆◆ Example 1: Graph y = sin x, with x in radians. Use the slopes of the tangents at points on that graph to sketch the graph of the derivative. Solution: We graph y = sin x as shown in Fig. 33-1(a) and the slopes as shown in Fig. 33-1(b). Note that the slope of the sine curve is zero at points A, B, C, and D, so the derivative curve must cross the x axis at points A′, B′, C′, and D′. We estimate the slope to be 1 at points E and F, which gives us points E′ and F′ on the derivative curve. Similarly, the slope is -1 at G, giving us point G′ on the derivative curve. We then note that the sine curve is rising from A to B and from C to D, so the derivative curve must be positive in those intervals. Similarly, the sine curve is falling from B to C, so the derivative curve is negative in this interval. Using all of this information, we sketch in the derivative curve. 33 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Find the derivatives of the sine and cosine functions. • Solve applied problems requiring derivatives of the trigonometric functions. • Find derivatives of the inverse trigonometric functions. • Find derivatives of logarithmic functions. • Find derivatives of exponential functions. In this chapter we extend our ability to take derivatives to include the trigonometric, logarith- mic, and exponential functions. This will enable us to solve a larger range of problems than was possible before. After learning the rules for taking derivatives of these functions, we apply them to problems quite similar to those in chapters 28 and 29; that is, tangents, related rates, maximum–minimum, and the rest.
eBook - PDFCalculus
Resequenced for Students in STEM
- David Dwyer, Mark Gruenwald(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
158 CHAPTER 3. THE DERIVATIVE Example 1 Finding a tangent line to a trigonometric function Find the equation of the line tangent to the graph of f (x) = 3 sin x +cos x at x = π 2 . Solution We first find the derivative of f . f (x) = 3 sin x + cos x f 0 (x) = 3 d dx [sin x] + d dx [cos x] = 3 cos x - sin x f 0 π 2 = 3 cos π 2 - sin π 2 = 3 · 0 - 1 = -1 So the tangent line has slope -1. Evaluating f at π 2 gives us the y-coordinate at the point of tangency. f π 2 = 3 sin π 2 + cos π 2 = 3 + 0 = 3 Since the tangent line has slope -1 and passes through ( π 2 , 3), its equation is y - 3 = -1 x - π 2 Example 2 Computing the second derivative of a product involving sine Compute d 2 dx 2 [x sin x]. Solution Applying the Product Rule, we have d dx [x sin x] = d dx [x] sin x + x d dx [sin x] = sin x + x cos x Differentiating again gives us d 2 dx 2 [x sin x] = d dx [sin x + x cos x] = cos x + [1 · cos x + x(- sin x)] = 2 cos x - x sin x Derivatives of Other Trigonometric Functions Differentiation formulas for the other trigonometric functions (tangent, cotangent, secant, and cosecant) can be found by expressing them in terms of sine or cosine and then applying the Quotient Rule. Example 3 Differentiating tan x Find a formula for d dx [tan x]. 3.5. TRIGONOMETRIC FUNCTIONS AND THEIR DERIVATIVES 159 Solution d dx [tan x] = d dx sin x cos x = cos x · d dx [sin x] - sin x · d dx [cos x] cos 2 x = cos x cos x - sin x(- sin x) cos 2 x = cos 2 x + sin 2 x cos 2 x = 1 cos 2 x = sec 2 x Example 4 Differentiating sec x Find a formula for d dx [sec x]. Solution d dx [sec x] = d dx 1 cos x = cos x · d dx [1] - 1 · d dx [cos x] cos 2 x = cos x · 0 - 1(- sin x) cos 2 x = sin x cos 2 x = 1 cos x sin x cos x = sec x tan x For convenience, we summarize the differentiation formulas for all six trigono- metric functions below. Proofs of the differentiation formulas for csc x and cot x are left for Exercises 21 and 22.
eBook - PDF- Ron Larson, Bruce Edwards(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
(The derivatives of the other four trigonometric functions are discussed in Section 3.3.) THEOREM 3.6 Derivatives of Sine and Cosine Functions d dx [sin x] = cos x d dx [cos x] = -sin x Proof Here is a proof of the first rule. (The proof of the second rule is left as an exercise [see Exercise 120].) In the proof, note the use of the trigonometric identity sin(x + Δx) = sin x cos Δx + cos x sin Δx. d dx [sin x] = lim Δxuni21920 sin(x + Δx) - sin x Δ x Definition of derivative = lim Δxuni21920 sin x cos Δ x + cos x sin Δ x - sin x Δ x = lim Δxuni21920 cos x sin Δx - (sin x)(1 - cos Δ x) Δ x = lim Δxuni21920 bracketleft.alt2 (cos x) parenleft.alt4 sin Δ x Δ x parenright.alt4 - (sin x) parenleft.alt4 1 - cos Δ x Δ x parenright.alt4bracketright.alt2 = (cos x) parenleft.alt4 lim Δxuni21920 sin Δ x Δ x parenright.alt4 - (sin x) parenleft.alt4 lim Δxuni21920 1 - cos Δ x Δ x parenright.alt4 = (cos x)(1) - (sin x)(0) = cos x This differentiation rule is shown graphically in Figure 3.18. Note that for each x, the slope of the sine curve is equal to the value of the cosine. Derivatives Involving Sines and Cosines See LarsonCalculus.com for an interactive version of this type of example. Function Derivative a. y = 2 sin x yuni2032 = 2 cos x b. y = sin x 2 = 1 2 sin x yuni2032 = 1 2 cos x = cos x 2 c. y = x + cos x yuni2032 = 1 - sin x d. y = cos x - π 3 sin x yuni2032 = -sin x - π 3 cos x TECHNOLOGY A graphing utility can provide insight into the interpretation of a derivative. For instance, Figure 3.19 shows the graphs of y = a sin x for a = 1 2 , 1, 3 2 , and 2. Estimate the slope of each graph at the point (0, 0). Then verify your estimates analytically by evaluating the derivative of each function when x = 0. FOR FURTHER INFORMATION For the outline of a geometric proof of the derivatives of the sine and cosine functions, see the article “The Spider’s Spacewalk Derivation of sinuni2032 and cosuni2032” by Tim Hesterberg in The College Mathematics Journal.
eBook - PDFCalculus
Early Transcendental Single Variable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
79 2 One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant. THE DERIVATIVE Many real-world phenomena involve changing quantities—the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a “derivative,” which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation. 2.1 TANGENT LINES AND RATES OF CHANGE In this section we will discuss three ideas: tangent lines to curves, the velocity of an object moving along a line, and the rate at which one variable changes relative to another. Our goal is to show how these seemingly unrelated ideas are, in actuality, closely linked. TANGENT LINES In Example 1 of Section 1.1 we used an informal argument to find the equation of a tangent line to a curve. However, at that stage in the text we did not have a precise definition of a tangent line. Now that limits have been defined precisely we can give a mathematical definition of the tangent line to a curve y = f (x) at a point P(x 0 , f (x 0 )) on the curve. As illustrated in Figure 2.1.1, the slope m PQ of the secant line through P and a second point Q(x, f (x)) on the graph of f is m PQ = f (x) − f (x 0 ) x − x 0 If we let x approach x 0 , then the point Q will move along the curve and approach the point P. Suppose the slope m PQ of the secant line through P and Q approaches a limit as x → x 0 . In that case we can take the value of the limit to be the slope m tan of the tangent line at P. Thus, we make the following definition. Figure 2.1.1
eBook - PDFCalculus
Single Variable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
59 2 One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant. THE DERIVATIVE Many real-world phenomena involve changing quantities—the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a “derivative,” which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation. 2.1 TANGENT LINES AND RATES OF CHANGE In this section we will discuss three ideas: tangent lines to curves, the velocity of an object moving along a line, and the rate at which one variable changes relative to another. Our goal is to show how these seemingly unrelated ideas are, in actuality, closely linked. TANGENT LINES In Example 1 of Section 1.1 we used an informal argument to find the equation of a tangent line to a curve. However, at that stage in the text we did not have a precise definition of a tangent line. Now that limits have been defined precisely we can give a mathematical definition of the tangent line to a curve y = f (x) at a point P(x 0 , f (x 0 )) on the curve. As illustrated in Figure 2.1.1, the slope m PQ of the secant line through P and a second point Q(x, f (x)) on the graph of f is m PQ = f (x) − f (x 0 ) x − x 0 If we let x approach x 0 , then the point Q will move along the curve and approach the point P. Suppose the slope m PQ of the secant line through P and Q approaches a limit as x → x 0 . In that case we can take the value of the limit to be the slope m tan of the tangent line at P. Thus, we make the following definition. Figure 2.1.1
eBook - PDF- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
THE DERIVATIVE 2 Henglein and Steets/Getty Images One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant. Many real-world phenomena involve changing quantities—the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a “derivative,” which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation. 2.1 TANGENT LINES AND RATES OF CHANGE In this section we will discuss three ideas: tangent lines to curves, the velocity of an object moving along a line, and the rate at which one variable changes relative to another. Our goal is to show how these seemingly unrelated ideas are, in actuality, closely linked. Tangent Lines In Example 1 of Section 1.1 we used an informal argument to find the equation of a tangent line to a curve. However, at that stage in the text we did not have a precise definition of a tangent line. Now that limits have been defined precisely we can give a mathematical definition of the tangent line to a curve y = f(x) at a point P(x 0 , f(x 0 )) on the curve. As illustrated in Figure 2.1.1, the slope m PQ of the secant line through P and a second point Q(x, f(x)) on the graph of f is m PQ = f(x) − f(x 0 ) x − x 0 Q Tangent line P y = f (x) x 0 x x - x 0 f (x) – f (x 0 ) x y f (x) f (x 0 ) Secant line ▴ Figure 2.1.1 If we let x approach x 0 , then the point Q will move along the curve and approach the point P. Suppose the slope m PQ of the secant line through P and Q approaches a limit as x → x 0 .
eBook - PDFThe Calculus Lifesaver
All the Tools You Need to Excel at Calculus
- Adrian Banner(Author)
- 2009(Publication Date)
- Princeton University Press(Publisher)
You could even write cot( x ) as the product cos( x ) csc( x ) and use the product rule. Whichever way you do it, you should get d dx cot( x ) = -csc 2 ( x ) . You should learn all six boxed formulas by heart. Notice that the three cofunctions (cos, csc, cot) all have minus signs in front of them, and the derivatives are the co-versions of the regular ones. For example, the derivative of sec( x ) is sec( x ) tan( x ), so throwing a “co” in front of everything and also putting in a minus sign, we get that the derivative of csc( x ) is -csc( x ) cot( x ). The same is true for cos and cot, remembering (in the case of cos) that co-co-sine is just the original sine function. By the way, what is the second derivative of f ( x ) = sin( x )? We know that f 0 ( x ) = cos( x ), and so f 00 ( x ) is the derivative of cos( x ), which we saw is -sin( x ). That is, d 2 dx 2 (sin( x )) = -sin( x ) . The second derivative of the function is just the negative of the original func-tion. The same is true for g ( x ) = cos( x ). This sort of thing doesn’t happen at all with (nonzero) polynomials, since the derivative of a polynomial is a new polynomial whose degree is one less than the original one. 7.2.1 Examples of differentiating trig functions Now that you have some more functions to differentiate, you’d better make sure you still know how to use the product rule, the quotient rule, and the chain rule. For example, how would you find the following derivatives: d dx ( x 2 sin( x )) , d dx sec( x ) x 5 and d dx (cot( x 3 ))? 144 • Trig Limits and Derivatives Let’s take them one at a time. If y = x 2 sin( x ), then we can write y = uv where u = x 2 and v = sin( x ). Now we just need to set up our table: u = x 2 v = sin( x ) du dx = 2 x dv dx = cos( x ) . Using the product rule (see Section 6.2.3 in the previous chapter), we get dy dx = v du dx + u dv dx = sin( x ) · (2 x ) + x 2 cos( x ) . This would normally be written as 2 x sin( x ) + x 2 cos( x ).
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