2.3 More Differentiation Patterns

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1 144 te derivative 2.3 More Differentiation Patterns Polynomials are very useful, but tey are not te only functions we need. Tis section uses te ideas of te two previous sections to develop tecniques for differentiating powers of functions, and to determine te derivatives of some particular functions tat occur often in applications: te trigonometric and exponential functions. As you focus on learning ow to differentiate different types and combinations of functions, it is important to remember wat derivatives are and wat tey measure. Calculators and computers are available to calculate derivatives. Part of your job as a professional will be to decide wic functions need to be differentiated and ow to use te resulting derivatives. You can succeed at tat only if you understand wat a derivative is and wat it measures. A Power Rule for Functions: D( f n (x If we apply te Product Rule to te product of a function wit itself, a familiar pattern emerges. D( f 2 D( f f f D( f + f D( f 2 f D( f D( f 3 D( f 2 f f 2 D( f + f D( f 2 f 2 D( f + f 2 f D( f 3 f 2 D( f D( f 4 D( f 3 f f 3 D( f + f D( f 3 f 3 D( f + f 3 f 2 D( f 4 f 3 D( f Practice 1. Wat is te pattern ere? Wat do you tink te results will be for D( f 5 and D( f 13? We could keep differentiating iger and iger powers of f (x by writing tem as products of lower powers of f (x and using te Product Rule, but te Power Rule For Functions guarantees tat te pattern we just saw for te small integer powers also works for all constant powers of functions. Te Power Rule for Functions is a special case of a more general teorem, te Cain Rule, wic we will examine in Section 2.4, so we will wait until ten to prove te Power Rule For Functions. Power Rule For Functions: If p is any constant ten D( f p (x p f p 1 (x D( f (x. Remember: sin 2 (x [sin(x] 2 Ceck tat you get te same answer by first expanding (x and ten taking te derivative. Example 1. Use te Power Rule for Functions to find: (a D((x d (b ( 2x + 3x 5 (c D(sin 2 (x Solution. (a To matc te pattern of te Power Rule for D((x 3 5 2, let f (x x 3 5 and p 2. Ten: D((x D( f p (x p f p 1 (x D( f (x 2(x D(x 3 5 2(x 3 5(3x 2 6x 2 (x 3 5

2 2.3 more differentiation patterns 145 (b To matc te pattern for d ( 2x + 3x 5 f (x 2x + 3x 5 and take p 2 1. Ten: d ( 2x + 3x 5 d ( f p (x p f p 1 (x 1 2 (2x + 3x5 1 2 d (2x + 3x5 d ( (2x + 3x 5 1 2, let d ( f (x 1 2 (2x + 3x5 1 2 (2 + 15x x4 2 2x + 3x 5 (c To matc te pattern for D(sin 2 (x, let f (x sin(x and p 2: D(sin 2 (x D( f p (x p f p 1 (x D( f (x 2 sin 1 (x D(sin(x 2 sin(x cos(x We could also rewrite tis last expression as sin(2x. Practice 2. Use te Power Rule for Functions to find: d ( (a (2x 5 π 2 (b D ( x + 7x 2 (c D(cos 2 (x Example 2. Use calculus to sow tat te line tangent to te circle x 2 + y 2 25 at te point (3, 4 as slope 3 4. Solution. Te top alf of te circle is te grap of f (x 25 x 2 so: f (x D ((25 x (25 x2 2 1 D(25 x 2 x 25 x 2 and f 3 (3 3. As a ceck, you can verify tat te slope of te radial line troug te center of te circle (0, 0 and te point (3, 4 as slope 4 3 and is perpendicular to te tangent line tat as a slope of 4 3. Derivatives of Trigonometric Functions We ave some general rules tat apply to any elementary combination of differentiable functions, but in order to use te rules we still need to know te derivatives of some basic functions. Here we will begin to add to te list of functions wose derivatives we know. We already know te derivatives of te sine and cosine functions, and eac of te oter four trigonometric functions is just a ratio involving sines or cosines. Using te Quotient Rule, we can easily differentiate te rest of te trigonometric functions. Teorem: D(tan(x sec 2 (x D(sec(x sec(x tan(x D(cot(x csc 2 (x D(csc(x csc(x cot(x

3 146 te derivative Proof. From trigonometry, we know tan(x sin(x cos(x, cot(x cos(x sin(x, sec(x 1 1 and csc(x. From calculus, we already know cos(x sin(x D(sin(x cos(x and D(cos(x sin(x. So: D(tan(x D ( sin(x cos(x cos(x D(sin(x sin(x D(cos(x (cos(x 2 cos(x cos(x sin(x( sin(x cos 2 (x cos2 (x + sin 2 (x cos 2 (x 1 cos 2 (x sec2 (x Similarly: ( 1 cos(x D(1 1 D(cos(x D(sec(x D cos(x (cos(x 2 cos(x 0 ( sin(x cos 2 (x sin(x cos 2 (x 1 cos(x sin(x sec(x tan(x cos(x Instead of te Quotient Rule, we could ave used te Power Rule to calculate D(sec(x D((cos(x 1. Practice 3. Use te Quotient Rule on f (x cot(x cos(x sin(x tat f (x csc 2 (x. to prove Practice 4. Prove tat D(csc(x csc(x cot(x. Te justification of tis result is very similar to te justification for D(sec(x. Practice 5. Find: (a D(x 5 tan(x (b d ( sec(t ( (c D cot(x x dt t Derivatives of Exponential Functions We can estimate te value of a derivative of an exponential function (a function of te form f (x a x were a > 0 by estimating te slope of te line tangent to te grap of suc a function, or we can numerically approximate tose slopes. Example 3. Estimate te value of te derivative of f (x 2 x at te point (0, 2 0 (0, 1 by approximating te slope of te line tangent to f (x 2 x at tat point. Solution. We can get estimates from te grap of f (x 2 x by carefully graping y 2 x for small values of x (so tat x is near 0, sketcing secant lines, and ten measuring te slopes of te secant lines (see margin figure.

4 2.3 more differentiation patterns 147 We can also estimate te slope numerically by using te definition of te derivative: f f (0 + f (0 2 (0 lim lim 0 0 lim and evaluating 2 1 for some very small values of. From te table below we can see tat f ( e 1 Practice 6. Fill in te table for 3 1 and sow tat te slope of te line tangent to g(x 3 x at (0, 1 is approximately At (0, 1, te slope of te tangent to y 2 x is less tan 1 and te slope of te tangent to y 3 x is sligtly greater tan 1 (see margin figure. You migt expect tat tere is a number b between 2 and 3 so tat te slope of te tangent to y b x is exactly 1. Indeed, tere is suc a number, e , wit lim 0 e 1 Te number e is irrational and plays a very important role in calculus and applications. We ave not proved tat tis number e wit te desired limit property actually exists, but if we assume it does, ten it becomes relatively straigtforward to calculate D(e x. 1 In fact, e is a transcendental number, wic means tat it is not te root of any algebraic equation. Don t worry we ll tie up some of tese loose ends in Capter 7. Teorem: D(e x e x Proof. Using te definition of te derivative: D(e x lim 0 e x+ e x lim 0 e x e e x lim e x e 1 lim e x e 1 lim e x 1 e x Te function f (x e x is its own derivative: f (x f (x.

5 148 te derivative Notice tat te limit property of e tat we assumed was true actually says tat for f (x e x, f (0 1. So knowing te derivative of f (x e x at a single point (x 0 allows us to determine its derivative at every oter point. Grapically: te eigt of f (x e x at any point and te slope of te tangent to f (x e x at tat point are te same: as te grap gets iger, its slope gets steeper. Example 4. Find: (a d ( t e t ( e x (b D (c D(e 5x dt sin(x Solution. (a Using te Product Rule wit f (t t and g(t e t : d ( t e t t D(e t + e t D(t t e t + e t 1 (t + 1e t dt (b Using te Quotient Rule wit f (x e x and g(x sin(x: ( e x D sin(x D(ex e x D(sin(x sin(x [sin(x] 2 sin(x ex e x ( cos(x sin 2 (x (c Using te Power Rule for Functions wit f (x e x and p 5: D((e x 5 5(e x 4 D(e x 5e 4x e x 5e 5x were we ave rewritten e 5x as (e x 5. Practice 7. Find: (a D(x 3 e x (b D((e x 3. Higer Derivatives: Derivatives of Derivatives Te derivative of a function f is a new function f and if tis new function is differentiable we can calculate te derivative of tis new function to get te derivative of te derivative of f, denoted by f and called te second derivative of f. For example, if f (x x 5 ten f (x 5x 4 and: f (x ( f (x (5x 4 20x 3 Definitions: Given a differentiable function f, te first derivative is f (x, te rate of cange of f. te second derivative is f (x ( f (x, te rate of cange of f. te tird derivative is f (x ( f (x, te rate of cange of f. For y f (x, we write f (x dy, so we can extend tat notation to write f (x d ( dy d2 y 2, f (x d ( d 2 y 2 d3 y and so on. 3

6 2.3 more differentiation patterns 149 Practice 8. Find f, f and f for f (x 3x 7, f (x sin(x and f (x x cos(x. If f (x represents te position of a particle at time x, ten v(x f (x will represent te velocity (rate of cange of te position of te particle and a(x v (x f (x will represent te acceleration (te rate of cange of te velocity of te particle. Example 5. Te eigt (in feet of a particle at time t seconds is given by t 3 4t 2 + 8t. Find te eigt, velocity and acceleration of te particle wen t 0, 1 and 2 seconds. Solution. f (t t 3 4t 2 + 8t so f (0 0 feet, f (1 5 feet and f (2 8 feet. Te velocity is given by v(t f (t 3t 2 8t + 8 so v(0 8 ft/sec, v(1 3 ft/sec and v(2 4 ft/sec. At eac of tese times te velocity is positive and te particle is moving upward (increasing in eigt. Te acceleration is a(t 6t 8 so a(0 8 ft/sec 2, a(1 2 ft/sec 2 and a(2 4 ft/sec 2. We will examine te geometric (grapical meaning of te second derivative in te next capter. A Really Bent Function In Section 1.2 we saw tat te oley function (x { 2 if x is a rational number 1 if x is an irrational number is discontinuous at every value of x, so (x is not differentiable anywere. We can create graps of continuous functions tat are not differentiable at several places just by putting corners at tose places, but ow many corners can a continuous function ave? How badly can a continuous function fail to be differentiable? In te mid-1800s, te German matematician Karl Weierstraß surprised and even socked te matematical world by creating a function tat was continuous everywere but differentiable nowere a function wose grap was everywere connected and everywere bent! He used tecniques we ave not investigated yet, but we can begin to see ow suc a function could be built. Start wit a function f 1 (see margin tat zigzags between te values 1 2 and 1 2 and as a corner at eac integer. Tis starting function f 1 is continuous everywere and is differentiable everywere except at te integers. Next create a list of functions f 2, f 3, f 4,..., eac of wic is sorter tan te previous one but wit many more corners tan te previous one. For example, we migt make f 2 zigzag between te

7 150 te derivative values 1 4 and 1 4 and ave corners at ± 1 2, ± 3 2, ± 5 2, etc.; f 3 zigzag between 1 9 and 1 9 and ave corners at ± 1 3, ± 2 3, ± 3 3 ±1, etc. If we add f 1 and f 2, we get a continuous function (because te sum of two continuous functions is continuous wit corners at 0, ± 1 2, ±1, ± 3 2,.... If we ten add f 3 to te previous sum, we get a new continuous function wit even more corners. If we continue adding te functions in our list indefinitely, te final result will be a continuous function tat is differentiable nowere. We aven t developed enoug matematics ere to precisely describe wat it means to add an infinite number of functions togeter or to verify tat te resulting function is nowere differentiable but we will. You can at least start to imagine wat a strange, totally bent function it must be. Until Weierstraß created is everywere continuous, nowere differentiable function, most matematicians tougt a continuous function could only be bad in a few places. Weierstraß function was (and is considered patological, a great example of ow bad someting can be. Te matematician Carles Hermite expressed a reaction sared by many wen tey first encounter te Weierstraß function: I turn away wit frigt and orror from tis lamentable evil of functions wic do not ave derivatives. Important Results Power Rule For Functions: D( f p (x p f p 1 (x D( f (x Derivatives of te Trigonometric Functions: D(sin(x cos(x D(tan(x sec 2 (x D(sec(x sec(x tan(x D(cos(x sin(x D(cot(x csc 2 (x D(csc(x csc(x cot(x Derivative of te Exponential Function: D(e x e x 2.3 Problems 1. Let f (1 2 and f (1 3. Find te values of eac of te following derivatives as x 2. (a D( f 2 (x (b D( f 2 (x (c D( f (x 2. Let f (2 2 and f (2 5. Find te values of eac of te following derivatives as x 2. (a D( f 2 (x (b D( f 3 (x (c D( f (x

8 2.3 more differentiation patterns For x 1 and x 3 estimate te values of f (x (wose grap appears below, f (x and d ( (a f 2 (x (b D ( f 3 (x (c D ( f 5 (x 4. For x 0 and x 2 estimate te values of f (x (wose grap appears above, f (x and (a D ( f 2 (x d ( (b f 3 d ( (x (c f 5 (x In Problems 5 10, find f (x. 5. f (x (2x f (x (6x x f (x x (3x f (x (2x (x f (x x 2 + 6x f (x x 5 (x A mass attaced to te end of a spring is at a eigt of (t 3 2 sin(t feet above te floor t seconds after it is released. (a Grap (t. (b At wat eigt is te mass wen it is released? (c How ig does above te floor and ow close to te floor does te mass ever get? (d Determine te eigt, velocity and acceleration at time t. (Be sure to include te correct units. (e Wy is tis an unrealistic model of te motion of a mass attaced to a real spring? 12. A mass attaced to a spring is at a eigt of (t 3 2 sin(t feet above te floor t seconds after it is t2 released. (a Grap (t. (b At wat eigt is te mass wen it is released? (c Determine te eigt and te velocity of te mass at time t. (d Wat appens to te eigt and te velocity of te mass a long time after it is released? 13. Te kinetic energy K of an object of mass m and velocity v is 1 2 mv2. (a Find te kinetic energy of an object wit mass m and eigt (t 5t feet at t 1 and t 2 seconds. (b Find te kinetic energy of an object wit mass m and eigt (t t 2 feet at t 1 and t 2 seconds. 14. An object of mass m is attaced to a spring and as eigt (t 3 + sin(t feet at time t seconds. (a Find te eigt and kinetic energy of te object wen t 1, 2 and 3 seconds. (b Find te rate of cange in te kinetic energy of te object wen t 1, 2 and 3 seconds. (c Can K ever be negative? Can dk ever be negative? dt Wy? In Problems 15 20, compute f (x. 15. f (x x sin(x 16. f (x sin 5 (x 17. f (x e x sec(x 18. f (x cos(x f (x e x + sin(x 20. f (x x 2 4x + 3 In 21 26, find an equation for te line tangent to te grap of y f (x at te given point. 21. f (x (x 5 7 at (4, f (x e x at (0, 1

9 152 te derivative 23. f (x 25 x 2 at (3, f (x sin 3 (x at (π, f (x (x a 5 at (a, f (x x cos 5 (x at (0, (a Find an equation for te line tangent to f (x e x at te point (3, e 3. (b Were will tis tangent line intersect te x- axis? (c Were will te tangent line to f (x e x at te point (p, e p intersect te x-axis? In Problems 28 33, calculate f and f. 43. Te function f (x defined as { x sin( 1 f (x x if x 0 0 if x 0 sown below is continuous at 0 because we can sow (using te Squeezing Teorem tat lim f (x 0 f (0 0 Is f differentiable at 0? To answer tis question, use te definition of f (0 and consider lim 0 f (0 + f (0 28. f (x 7x 2 + 5x f (x cos(x 30. f (x sin(x 31. f (x x 2 sin(x 32. f (x x sin(x 33. f (x e x cos(x 34. Calculate te first 8 derivatives of f (x sin(x. Wat is te pattern? Wat is te 208t derivative of sin(x? 35. Wat will te second derivative of a quadratic polynomial be? Te tird derivative? Te fourt derivative? 36. Wat will te tird derivative of a cubic polynomial be? Te fourt derivative? 37. Wat can you say about te n-t and (n + 1-st derivatives of a polynomial of degree n? In Problems 38 42, you are given f. Find a function f wit te given derivative. 38. f (x 4x f (x 5e x 40. f (x 3 sin 2 (x cos(x 41. f (x 5(1 + e x 4 e x 42. f (x e x + sin(x 44. Te function f (x defined as { x f (x 2 sin( 1 x if x 0 0 if x 0 (sown at te top of te next page is continuous at 0 because we can sow (using te Squeezing Teorem tat lim f (x 0 f (0 0 Is f differentiable at 0? To answer tis question, use te definition of f (0 and consider lim 0 f (0 + f (0

10 2.3 more differentiation patterns Define n! to be te product of all positive integers from 1 troug n. For example, 2! 1 2 2, 3! and 4! (a Calculate te value of te sums: s ! s ! + 1 2! s ! + 1 2! + 1 3! s ! + 1 2! + 1 3! + 1 4! Te number e appears in a variety of unusual situations. Problems illustrate a few of tese. 45. Use your calculator to examine te values of f (x (1 + 1 x x wen x is relatively large (for example, x 100, 1000 and 10, 000. Try some oter large values for x. If x is large, te value of f (x is close to wat number? 46. If you put $1 into a bank account tat pays 1% interest per year and compounds te interest x times a year, ten after one year you will ave ( x x dollars in te account. (a How muc money will you ave after one year if te bank calculates te interest once a year? (b How muc money will you ave after one year if te bank calculates te interest twice a year? (c How muc money will you ave after one year if te bank calculates te interest 365 times a year? (d How does your answer to part (c compare wit e 0.01? s ! + 1 2! + 1 3! + 1 4! + 1 5! s ! + 1 2! + 1 3! + 1 4! + 1 5! + 1 6! (b Wat value do te sums in part (a seem to be approacing? (c Calculate s 7 and s If it is late at nigt and you are tired of studying calculus, try te following experiment wit a friend. Take te 2 troug 10 of earts from a regular deck of cards and suffle tese nine cards well. Have your friend do te same wit te 2 troug 10 of spades. Now compare your cards one at a time. If tere is a matc, for example you bot play a 5, ten te game is over and you win. If you make it troug te entire nine cards wit no matc, ten your friend wins. If you play te game many times, ten te ratio: total number of games played number of times your friend wins will be approximately equal to e. 2.3 Practice Answers 1. Te pattern is D( f n (x n f n 1 (x D( f (x: D( f 5 (x 5 f 4 (x D( f and D( f 13 (x 13 f 12 D( f d 2. (2x5 π 2 2(2x 5 π 1 D(2x 5 π 2(2x 5 π(10x 4 40x 9 20πx 4 ( D (x + 7x (x + 7x2 2 1 D(x + 7x x 2 ( x + 7x 2 D (cos(x 4 4(cos(x 3 D(cos(x 4(cos(x 3 ( sin(x 4 cos 3 (x sin(x

11 154 te derivative 3. Mimicking te proof for te derivative of tan(x: ( cos(x sin(x D(cos(x cos(x D(sin(x D sin(x (sin(x 2 sin(x( sin(x cos(x(cos(x sin 2 (x (sin2 (x + cos 2 (x sin 2 (x 1 sin 2 (x csc2 (x 4. Mimicking te proof for te derivative of sec(x: ( 1 sin(x D(1 1 D(sin(x D(csc(x D sin(x sin 2 (x sin(x 0 cos(x sin 2 1 (x sin(x cos(x cot(x csc(x sin(x 5. D(x ( 5 tan(x x 5 D(tan(x + tan(x D(x 5 x 5 sec 2 (x + tan(x(5x 4 d sec(t t D(sec(t sec(t D(t t sec(t tan(t sec(t dt t t 2 t 2 ( D (cot(x x (cot(x x 1 2 D(cot(x x 6. Filling in values for bot 3 x and e x : 1 2 (cot(x x 1 2 ( csc 2 (x 1 csc2 (x 1 2 cot(x x e D(x ( 3 e x x 3 D(e x + e x D(x 3 x 3 e x + e x 3x 2 x 2 e x (x + 3 D (e x 3 3 (e x 2 D(e x 3e 2x e x 3e 3x 8. f (x 3x 7 f (x 21x 6 f (x 126x 5 f (x 630x 4 f (x sin(x f (x cos(x f (x sin(x f (x cos(x f (x x cos(x f (x x sin(x + cos(x f (x x cos(x 2 sin(x f (x x sin(x 3 cos(x