6 Computing Derivatives the Quick and Easy Way

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1 Jay Daigle Occiental College Mat 4: Calculus Experience 6 Computing Derivatives te Quick an Easy Way In te previous section we talke about wat te erivative is, an we compute several examples, an ten we got quite tire of computing tose examples over an over. In tis section we ll come up wit some tecniques to make computation of erivatives easier. It correspons rougly to sections 2.3 an 2.4 in your textbook. 6. Basics: Te simple rules. If c is a constant an fx c ten f x 0. f x fx + fx c c 0 0. Conceptually, a constant function never canges, so te rate of cange is 0. Geometrically, a constant function is a orizontal line; tus we tink of te slope everywere as being 0. Example If fx x, ten f x. f x fx + fx x + x. Conceptually, if we ave te ientity function, ten wenever we cange te input ten te output soul cange by exactly te same amount. Tus te rate of cange is. Geometrically, tis is a line wit slope. 3. If c is a constant an g is a function an fx c gx, ten f x cg x. f x cgx + cgx c gx + gx c g x. Conceptually, if canging x by a bit canges gx by a certain amount, ten it will cange 2gx by twice tat amount multiplying by a scalar soul just cange te rate of cange by te same amount everywere. Geometrically, multiplying by a constant is just stretcing vertically an all te slopes will be stretce by tat same amount. ttp://jayaigle.net/teacing/courses/208-spring-4/ 59

2 Jay Daigle Occiental College Mat 4: Calculus Experience Example 6.2. If fx 5x ten f x 5 x 5 x If f an g are functions ten f + g x f x + g x. f + g fx + + gx + fx gx x fx + fx gx + gx + f x + g x. Conceptually, if canging te input by a bit canges f by a certain amount an g by a ifferent amount, ten it canges f + g by te sum of tose two amounts figure out ow muc it canges eac part an ten a tem togeter to fin out ow muc it canges te wole. Geometrically, if we a two functions togeter it s just like stacking tem on top of one anoter, so te slope at any point will be te sum of te slopes. Example 6.3. Let fx 3x 7. Ten f x 3x 7 3x 0 3. Tis rule is really important but so far we can t o muc wit it we on t ave quite enoug rules yet. 5. Power Rule If fx x n were n is a positive integer, ten f x nx n. In fact, if gx x r an r is any real number, ten g x rx r. We ll only prove tis for integers, using te ifference-of-nt-powers rule. f z n x n x z x z x z xz n + z n 2 x + + zx n 2 + x n z x z x z x z n + z n 2 x + + zx n 2 + x n x n + + x n nx n. Now tat we ave tis, we can compute all sorts of erivatives. Example 6.4. x 2 + 2x + 0 2x. x x /2 2 x /2 2 x. 3 x x /3 3 x 2/3 3 3 x 2. 3 x + x x + 5x ttp://jayaigle.net/teacing/courses/208-spring-4/ 60

3 Jay Daigle Occiental College Mat 4: Calculus Experience 6. Prouct Rule If f an g are functions ten fg x f xgx + fxg x. Conceptually, we sort of know tis alreay; if we a a bit on to f an a bit on to g, ten we get f + f g + g fg + fg + gf + g f, an in te it we can treat g f as being zero. So tis is te same as multiplying te bit we a to g wit f, an multiplying te bit we a to f wit g, an ten aing te two. Example x 2x 3x 2 5x + 2 6x 5. Alternatively, 3x 2x 3x 2 x +3x 2x 3 x + 3x 2 6x 5. Tis rule isn t terribly important as long as we re only working wit rational functions. Once we inclue anyting else, like trig functions, it is critical. Remark 6.6. We can get te power rule from te prouct rule instea of trying to get it irectly. 7. Quotient Rule: If f an g are functions ten f/g x fx+ fx gx+ gx f/g x f xgx fxg x gx 2. fx + gx fxgx + gx + gx fx + gx fxgx + fxgx fxgx + gx + gx fx + gx fxgx + gx + gx gx 2 fx + fx gx + gx gx fx f xgx fxg x gx 2 x Example 6.7. x 2 x 3 2x 3 + 3x 4. x 3 fxgx fxgx + Alternatively, x x x 3 x 3x 2 x3 3x 3 + 3x 2 2x 3 + 3x 4. x 3 x 6 x x 2 + 3x 3 5x 2 + 3x3 5x 9 5x x 9 3 5x 3 5x 2 3 5x 2 3 5x 2 ttp://jayaigle.net/teacing/courses/208-spring-4/ 6

4 Jay Daigle Occiental College Mat 4: Calculus Experience 6.2 Trigonometric erivatives We cannot neglect te trigonometric functions no matter ow muc we migt wis to on occasion. All of te rules for trigonometric erivatives rely on wat are known as te angle aition formulas: sina + b sina cosb + cosa sinb cosa + b cosa cosb sina sinb. Note: you probably won t ever nee to know tese formulas again in tis class. But I will nee tem for anoter page or so of tese notes. Using tis we can compute. sinx sinx + sinx sinx cos + sin cosx sinx sin cosx sinxcos + sin cos cosx + sinx cos 2 cosx + sinx cos + sin 2 cosx sinx cosx sinx cos + sin cos + cosx sinx 0 cosx. sin 2. A similar argument sows tat cosx sinx. Furter using te prouct an quotient rules, we observe tat tanx cotx secx sin x cos 2 x + sin 2 x cos x cos 2 x cos x sin 2 x cos 2 x sin x sin 2 x 0 + sin x cos x cos 2 x sin x cos x cos x cos 2 x sec2 x sin 2 x csc2 x secx tanx ttp://jayaigle.net/teacing/courses/208-spring-4/ 62

5 Jay Daigle Occiental College Mat 4: Calculus Experience cscx 0 cosx sin x sin 2 x cos x sin x sin x cscx cotx. Remember tat as long as you know te erivatives of sin an cos you can always compute tese four erivatives wenever you nee tem. Example If ft 3 sin t + cos t, ten f t 3 cos t sin t. 2. Fin te tangent line to y 6 cos x at π/3, 3. We see tat y 6 sin x, an tus wen x π/3 we ave y 3 3. Recalling tat te equation of our line is y mx x 0 + fx 0, we ave te equation y 3 3x π/ IF gθ θ sin θ + cos θ, ten θ g θ sin θ + θ cos θ + θ sin θ cos θ θ If x x 2 tan x, ten x 2 tan x + x sec2 x 2 tan x We can also compute secon erivatives. sin x sin x. cos x cos x. tan x sec x sec x sec x tan x sec x + sec x tan x sec x 2 sec 2 x tan x. 6.3 Te Cain Rule To start wit an example, suppose gx sin x 2. Ten g x sin xsin x cos x sin x + cos x sin x 2 sin x cos x. Remembering tat x 2 2x, we notice tat tis looks suggestive. It also leas us to ask wat appens wen we buil up functions by composition, tat is, plugging one function into anoter, as we ave ere. If we want to freely buil complex functions from simple ones, we nee to be able to combine tem in cains. Remember tat we efine te function f g by f gx fgx; we take our input x, plug it into g, an ten take te output gx an plug it into f. We can see ow tis is useful in two ifferent ways. First, as we saw earlier, it lets us buil up functions. ttp://jayaigle.net/teacing/courses/208-spring-4/ 63

6 Jay Daigle Occiental College Mat 4: Calculus Experience. x + 2 f gx were gx x + an fx x x f gx were gx x 2 + an fx x sin 2 x f gx were gx sin x an fx x 2. Secon, sometimes composition of functions really is te best way to escribe wat s going on, especially wen you ave a causal cain were one process causes a secon wic causes a tir. For instance, suppose you re riving up a mountain at 2 km/r, an te temperature rops 6.5 C per kilometer of altitue. You can tink about your temperature as a function of your eigt, wic is itself a function of te time; ten te numbers I gave you are te rates of cange, or erivatives, of eac function. It s not tat ar to convince yourself tat you ll get coler by about 3 C per our. Does tis work in general? Proposition 6.9 Cain Rule. Suppose f an g are functions, suc tat g is ifferentiable at a an f is ifferentiable at ga. Ten f g a f ga g a. Proof. f g f ga + f ga a fga + fga ga + ga fga + fga ga + ga f ga g a. ga + ga ga + ga Remark Wen we write f gx, we mean te function f evaluate at te point gx, or in oter wors, te erivative of f at te point gx. 2. It can be elpful as a way of remembering te cain rule tat f g x f g g g x. Don t take tis too seriously as actively meaning anyting, since it only sort of oes, but it s quite elpful for te memory. ttp://jayaigle.net/teacing/courses/208-spring-4/ 64

7 Jay Daigle Occiental College Mat 4: Calculus Experience Example 6... x + 2 f gx were gx x + an fx x 2. Ten f x 2x an g x, so f g x f gx g x 2gx 2x + 2x + 2. Sanity ceck: f g x x 2 + 2x + 2x x f gx were gx x 2 + an fx x 2. Ten f 2x, g 2x, an f g x f gx g x 2gx 2x 2x 2 + 2x 4x 3 + 4x. Sanity ceck: f g x x 4 + 2x 2 + 4x 3 + 4x. 3. sin 2 x f gx were gx sin x an fx x 2. Ten f x 2x, g x cos x, an we ave f g x 2gx cos x 2sin x cos x. 4. cos3x f gx were fx cosx an gx 3x. Ten f x sinx an g x 3 an f g x sin3x sinx 2 f gx were fx sinx an gx x 2. Ten f x cos x, g x 2x, an f g x cosgx 2x 2x cosx If fx is any function, ten we can write fx r as g fx were gx x r. Ten x fxr g f x rfx r f x. 7. Te erivative of sec5x is sec5x tan5x5. 8. Wat is te erivative of 3 x 4 2x+ using te cain rule, we ave x? We can view tis as x4 2x + /3, an 3 x4 2x + 3 x4 2x + 4/3 4x Wat is te erivative of sec 2 x? By te cain rule tis is 2 secx sec x 2 secx secx tanx 2 sec 2 x tanx. ttp://jayaigle.net/teacing/courses/208-spring-4/ 65

8 Jay Daigle Occiental College Mat 4: Calculus Experience 0. Wat is te erivative of tan 4 x? We get 4 tan 3 x sec x 4 tan 3 x secx tanx 4 tan 4 x secx.. Sometimes we ave to nest te cain rule. Wat is te erivative of x 3 + x 2 +? We can pull tis apart slowly. x x 3 + x x3 + x 2 + /2 2 x 3 + x 2 + 3x2 + 2x 2 x x 3 + x Barnburners x x 3 + x 2 + 3x x2 + /2 x x2 + As we saw, te cain rule can stack, or cain togeter. As functions get more complicate we will ave to use multiple applications of te prouct rule, quotient rule, an cain rule to pull our erivative apart. Poll Question Fin x secx2 + x 3 +. x secx2 + x 3 + secx 2 + x 3 + tanx 2 + x 3 + x 2 + x 3 + secx 2 + x 3 + tanx 2 + x 3 + 2x + 2 x3 + /2 3x 2 Poll Question Fin sinx 2 + sin 2 x x x 2 + sinx 2 + sin 2 x x x 2 + sinx2 + sin 2 x x 2 + 2xsinx 2 + sin 2 x x cosx2 2x + 2 sinx cosxx 2 + 2xsinx 2 + sin 2 x x We can keep going wit increasingly complicate problems, basically until we get bore. Tese are really goo practice for making sure you unerstan ow te rules fit togeter. Example 6.2. Fin x + x cos x + 2 ttp://jayaigle.net/teacing/courses/208-spring-4/ 66

9 Jay Daigle Occiental College Mat 4: Calculus Experience x + x cos x + /2 x + x cos x + 2 cos x + 2 /2 x + 2 x /2 cos x + 2 2cos x + sin x x + 2 cos x + 2 cos x + 4 Example 6.3. Calculate sin 2 x 2 + x + x 3 2 x cos x tanx /3 ttp://jayaigle.net/teacing/courses/208-spring-4/ 67