Chapter 3. Differentiation 3.2 Differentiation Rules for Polynomials, Exponentials, Products and Quotients

Transcription

1 3.2 Dfferetato Rules 1 Capter 3. Dfferetato 3.2 Dfferetato Rules for Polyomals, Expoetals, Proucts a Quotets Rule 1. Dervatve of a Costat Fucto. If f as te costat value f(x) = c, te f x = [c] = 0. x Proof. From te efto: f (x) 0 f(x + ) f(x) o c c 0 0 = 0. Rule 2. Power Rule for Postve Itegers If s a postve teger, te x [x ] = x 1. Note. Before we preset te proof of te Power Rule, we trouce te Bomal Teorem.

2 3.2 Dfferetato Rules 2 Teorem. Bomal Teorem Let a a b be real umbers a let be a postve teger. Te (a + b) = a + a 1 b + = =0 a b ( 1) a 2 b ab 1 + b 2 were =! ( )!! a! = ()( 1)( 2) (3)(2)(1). Note. We ca prove te Bomal Teorem usg Matematcal Iucto. Proof of te Power Rule. By efto, f f(x + ) f(x) (x) 0 (x + ) x =0 x + =1 x x x x

3 3.2 Dfferetato Rules =1 =1 =1 x 0 x 1 + =2 = x 1. x 1 x 1 x 1 Note. See page 157 for a proof of te Power Rule tat oes t (explctly) use te Bomal Teorem. Rule 3. Costat Multple Rule If u s a fferetable fucto of x, a c s a costat, te [cu] = cu x x.

4 3.2 Dfferetato Rules 4 Rule 4. Dervatve Sum Rule If u a v are fferetable fuctos of x, te ter sum u + v s fferetable at every pot were u a v are bot fferetable. At suc pots, u [u + v] = x x + v x. Note. Te proofs of Rules 3 a 4 follow from te correspog rules for lmts (amely, te Costat Multple Rule a te Sum Rule, respectvely). Corollary. If P(x) = a x + a 1 x a 2 x 2 + a 1 x + a 0, te P (x) = a x 1 + ( 1)a 1 x a 2 x + a 1. Note. We ow fferetate a expoetal fucto f(x) = a x were a > 0. By efto, f (x) 0 f(x + ) f(x) 0 a x+ a x 0 a x a a x

5 3.2 Dfferetato Rules 5 a xa 1 0 = a x a 1 lm 0. a 1 We clam wtout justfcato tat lm exsts a s some 0 umber L a epeet o a. (For a clea scusso of ts result, see sectos 7.2 a 7.3 of Tomas Calculus, Staar 11t Eto otes are avalable ole at ttp:// otes11.tm. Tere s a verso ts text secto 7.1.) Wt x = 0, we ave f (0) = a 0 a 1 a 1 lm = L a. We wll see te precse 0 0 value of L a secto 3.7. Now f (0) s te slope of te grap of y = a x at x = 0. Motvate by Fgure 3.11, we see tat tere s a value of a somewere betwee 2 a 3 suc tat ts slope s 0. Fgure 3.11, page 161

6 3.2 Dfferetato Rules 6 We efe e to be te umber for wc te slope of te le taget to y = e x e 1 s m = 1 at x = 0. Tat s, we efe e suc tat lm = 0 1. Oe ca eterme umercally (for a tecque, see page 220) tat e Wat s atural about te atural expoetal fucto e x s a calculus property a fferetato property. Teorem. Dervatve of te Natural Expoetal Fucto. x [ex ] = e x. Example. Dfferetate f(x) = x + 5e x. Note. We ca also calculate ger orer ervatves: y = x [y ], y = x [y ], y (4) = x [y ],..., y () = x [y( 1) ]. Example. Page 167 umber 30.

7 3.2 Dfferetato Rules 7 Rule 5. Dervatve Prouct Rule If u a v are fferetable at x, te so s ter prouct uv, a u [uv] = x x v + uv x = [u ]v + u[v ]. Proof. By efto we ave: u(x + )v(x + ) u(x)v(x) [uv] x 0 u(x + )v(x + ) u(x + )v(x) + u(x + )v(x) u(x)v(x) 0 v(x + ) v(x) u(x + ) u(x) u(x + ) + v(x) 0 v(x + ) v(x) u(x + ) lm 0 0 = u(x)[v (x)] + [u (x)]v(x). + v(x) lm 0 u(x + ) u(x) were lm u(x + ) = u(x) sce u s cotuous at x by Teorem 1 of 0 secto 2.1. Example. Dfferetate f(x) = (4x 3 5x 2 + 4)(7x 2 x).

8 3.2 Dfferetato Rules 8 Rule 6. Dervatve Quotet Rule If u a v are fferetable at x a f v(x) 0, te te quotet u/v s fferetable at x, a x [ ] u v = u x v uv x v 2 = [u ]v u[v ] v 2. Proof. By efto we ave: x [ ] u v 0 u(x+) v(x+) u(x) v(x) 0 v(x)u(x + ) u(x)v(x + ) v(x + )v(x) v(x)u(x + ) v(x)u(x) + v(x)u(x) u(x)v(x + ) 0 v(x + )v(x) v(x) u(x+) u(x) u(x) v(x+) v(x) x 0 v(x + )v(x) 0 v(x) u(x+) u(x) = v(x) lm 0 u(x+) u(x) = v(x)u (x) u(x)v (x). v 2 (x) lm 0 u(x) v(x+) v(x) lm 0 v(x + )v(x) u(x) lm 0 v(x+) v(x) v(x) lm 0 v(x + ) Example. Page 167 umber 20.

9 3.2 Dfferetato Rules 9 Rule 7. Power Rule for Negatve Itegers If s a egatve teger a x 0, te x [x ] = x 1. Proof. If s a egatve teger, te s a postve teger a we ca use te Power Rule for Noegatve Itegers to fferetate x. So x [x ] = = 1 x x x [1] (x ) (1) x [x ] (x ) 2 = [0](x ) (1)[( )x 1 ] x 2 = x 1. Note. We ave ow establse tat x [x ] = x 1 for all tegers. We wll evetually see tat ts s te way x s fferetate for all real umbers, but we ave ot eve efe wat t meas to rase a real umber to a rratoal umber! We wll take care of ts we we efe te atural logartm a expoetal fuctos. Example. Page 169 umber 56, page 167 umber 34. Note. We wll follow my square brackets otato as escrbe te aout.