Derivative of Trigonometric Functions

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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  = = - d dx x x d dx x x sin( ) cos( ), cos( ) sin( ) (8.22) The derivatives of the remaining four trigonometric ratios can be obtained by means of the reciprocal trigonometric identities and the quotient and product rules. For example, d dx x d dx x x x x x x x x x x x x tan( ) sin( ) cos( ) cos( ) cos( ) sin( ) ( sin( )) cos ( ) cos ( ) sin ( ) cos ( ) 1 cos ( ) sec ( ). 2 2 2 2 2 2 = = ⋅ - ⋅ - = + = = The proofs of the remaining trigonometric ratios are left as exercises. It is well worthwhile memorizing their derivatives given in table 8.1. TABLE 8.1. Derivatives of trigonometric ratios d dx x x sin( ) cos( ) = d dx x x cos( ) sin( ) = - d dx x x tan( ) sec 2 = d dx x x cot( ) csc 2 = - d dx x x x sec( ) sec( ) tan( ) = d dx x x x csc( ) csc( ) cot( ) = - The following examples demonstrate the product rule combined with the derivatives of trigo- nometric functions. EXAMPLE 8.6.1. (i) d dx x x x x x x x x sin( ) sin( ) cos( ) sin( ) cos( ) = + = + (ii) d dx x x x x x x x sec( ) 2 sec( ) sec( ) tan( ) 2 2 = +  8.7 SOME BASIC APPLICATIONS OF CALCULUS By means of the examples below, we demonstrate how the techniques of calculus can be used to solve many kinds of practical problems. In the first example, we introduce the important notion of instantaneous rate of change and explain how it relates to the derivative of a time–displacement function of an object in motion. The general study or science of motion is called dynamics, and, because of this relationship of the Differential Calculus • 263 derivative to instantaneous change, calculus is an essential tool in dynamics. A more general and sophisticated expression for a moving body than a simple time–displacement function is a differen- tial equation. Students often ask why they need to learn calculus. The best answer to this question is that calculus is the foundation for the mathematics of differential equations.

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  Calculus

  Late Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2021(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

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  Calculus

  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

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  Section 7.2: Derivatives Involving Trig Functions • 141 0. Finally, as you can see, the function isn’t continuous at x = 0 since it isn’t defined there. However, if we define the function g by g ( x ) = sin( x ) /x if x 6 = 0 and g (0) = 1, then we have effectively filled in the open circle at (0 , 1) in the above picture, and the function g is continuous. 7.2 Derivatives Involving Trig Functions Now, time to differentiate some functions. Let’s start off by differentiating sin( x ) with respect to x . To do this, we’re going to use two of the limits from Section 7.1.2 above: lim h → 0 sin( h ) h = 1 and lim h → 0 1 -cos( h ) h = 0 . (OK, so I changed x to h , but no matter—the h is a dummy variable anyway and could be replaced by any letter at all.) Anyway, with f ( x ) = sin( x ), let’s differentiate: f 0 ( x ) = lim h → 0 f ( x + h ) -f ( x ) h = lim h → 0 sin( x + h ) -sin( x ) h . Now what? Well, you should remember the formula sin( A + B ) = sin( A ) cos( B ) + cos( A ) sin( B ); if not, you’d better look at Chapter 2 again. Anyway, we want to replace A by x and B by h , so we have sin( x + h ) = sin( x ) cos( h ) + cos( x ) sin( h ) . Inserting this in the above limit, we get f 0 ( x ) = lim h → 0 sin( x ) cos( h ) + cos( x ) sin( h ) -sin( x ) h . All that’s left is to group the terms a little differently and do a bit of factoring; we get f 0 ( x ) = lim h → 0 sin( x )(cos( h ) -1) + cos( x ) sin( h ) h = lim h → 0 sin( x ) cos( h ) -1 h + cos( x ) sin( h ) h . Notice that we separated as much x -stuff as we could from h -stuff. Now we actually have to take the limit as h → 0 (not as x → 0!). Using the two limits from the beginning of this section, we get f 0 ( x ) = sin( x ) × 0 + cos( x ) × 1 = cos( x ) . That is, the derivative of f ( x ) = sin( x ) is f 0 ( x ) = cos( x ), or in other words, d dx sin( x ) = cos( x ) . 142 • Trig Limits and Derivatives Now you should try to repeat the argument but this time with f ( x ) = cos( x ).

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  A Course of Mathematical Analysis

  International Series of Monographs on Pure and Applied Mathematics

  • A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  We can say briefly that: The derivative of a composite function is equal to the product of the derivatives of the component functions. Example. l-(l-x 2 ) 2 y = l + ( l -# 2 ) 2 The function is obtained by raising to the third power a linear rational function of the square (1 — x 2 ) 2 . We distinguish the inter-mediate arguments as: y == u 3 , U = — , V = W 2 , 1 -f v w — 1 — x 2 . We find by using the rule: y ' = ( U *y u'v'w' = ( ttS )'(y—^) (w 2 )' (1 -x 2 Y = 3u2 /i 2 % ^(-2x) = 24u 2 XW (1 + v) 2 v (1 + v) 2 ' Substitution for w, v and u by their expressions in terms of x gives 1 -( l -a 2 ) 2 l 2 x(l-x 2 ) y f = 2± l + ( l -' z 2 ) 2 [l+(l-a; 2 ) 2 ] 2 As experience is gained in differentiation it will become unnecessary to introduce special notation for the intermediate arguments. Dif-ferentiation of functions is usually carried out by distinguishing only mentally the elementary branches leading from y to x. 48. Derivatives of the basic elementary functions. We shall now find the derivatives of all the basic elementary functions; we start with the trigonometric functions. DERIVATIVES AND DIFFERENTIALS 151 I. THE DERIVATIVES OF THE TRIGONOMETRIC FUNCTIONS. Let y — sin#. Let x receive the increment A x ; we shall write for brevity A x = h. Now: Λ -/ , Ζ Λ · ^ . x+h-x x + h + x Δ y = sin (x + h) — sin x = 2 sin cos »-= 2 sin Consequently, and h cos (a; Ay h + = §)· . h s i n -h ~2 cos A K) sin-i/ = lim —— = hm — lim cos [x + — . The first factor is equal to unity by what was proved in Sec. 40, whilst lim cosl x + — ] = cos (x + 0) = cos x , Λ->0 2 / since cos a; is a continuous function. Altogether, y' = (sin a:)' = cos a:. The derivatives of the remaining trigonometric functions are easily found by the rules for differentiation. Let us take y = cosa Since cos a; = sin (a: + π), we find on putting y' = (sin u)' u' = cos u · 1 = cosI x + — ) = — sin x, i.e. y' = (cosa;)' = — sina;.

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