The Derivative of a Constant is Zero

Transcription

1 Sme Simple Algrihms fr Calculaing Derivaives The Derivaive f a Cnsan is Zer Suppse we are l ha x x where x is a cnsan an x represens he psiin f an bjec n a sraigh line pah, in her wrs, he isance ha he bjec is in frn f a sar line. The erivaive f x wih respec which can als be wrien x is hen jus he erivaive f x wih respec. Nw accring he mahemaicians, he erivaive f a cnsan is zer, s we have: Des his make sense? This whle iscussin abu erivaives is relevan he suy f min because he velciy f an bjec is he erivaive f is psiin wih respec ime: v S wha we are saying nw is ha if x x (where x is a cnsan) hen v 0. Well x x means ha he psiin f he bjec is n changing. S if we are alking abu a car, fr insance, hen we mus be alking abu a parke car, an YES i es make sense fr 0, ha is fr v 0, because he velciy f a parke car is inee zer. 0 An algrihm is a sequence f seps. When yu learne hw lng ivisin, yu were learning execue an algrihm. The erm is usually use in cmpuer science where he gal is fen wrie an algrihm in a language ha a cmpuer can unersan. Bu he erm is n limie cmpuer science. In fac, yu have been execuing algrihms yurself ever since yu firs learne ha ne plus ne is w.

2 The Derivaive f Wih Respec Is. Cnsier he funcin f ( ) which is be rea f f equals. Accring he mahemaicians, if f hen f Des his make sense? When yu ake he erivaive f smehing wih respec ime, f yu are suppse ge he rae f change f ha smehing. If f, hen is he rae f change f he spwach reaing. Our resul,, wih n unis, can be inerpree as s minue hur r r, any ne f which is inee he rae a which he spwach s minue hur reaing changes. The Derivaive f a Cnsan Times a Funcin is he Cnsan Times he Derivaive f he Funcin. The erivaive f x( ) vf ( ) which is be rea x f equals v-sub- imes f f ) where v is a cnsan is f ( v ) v v bu we alreay esablishe ha he erivaive f wih respec is. S, v Tha s a mahemaical resul. Nw les lk a he physics 3. If x( ) v wih v being cnsan, hen: (a) x 0 when 0 meaning he bjec is a he sar line a ime zer (he insan when he spwach is sare). I chse use a funcin name her han x here because I am using x fr psiin an fr spwach reaing, an, he unis n allw us se x equal. 3 By he physics we mean he way in which he paricular aspec f naure uner cnsierain is r behaves.

3 x (b) Ne ha if we slve his fr v we ge v which can be wrien as x 0 x v which is ur expressin fr he average velciy. Since i was 0 sipulae ha v is a cnsan, we mus ge he same value fr average velciy n maer wha (x, ) pair we pick. This means ha he velciy is a cnsan a he value f v. S when x( ) v he bjec is mving a cnsan velciy v. Velciy is he rae f change f psiin bu ha is exacly wha we mean by s ur mahemaical resul ha v is in agreemen wih he physics. Geing back he iea ha he erivaive f a cnsan imes a funcin f wih respec is jus he cnsan imes he erivaive f he funcin, les lk a anher example. Suppse where a is a cnsan. If a is a cnsan hen frm x ( ) a a x( ) cnsan f ( ) is, f curse, a cnsan s x() has he (which is be rea x f equals a cnsan imes f f ) where f ( ) he rule: cnsan an, in he case a han, where he cnsan is have a a f which invlves aking he erivaive f a pwer f.. Accring an f, ha is, x ( ) a, we 3

4 The Derivaive f n Wih Respec is n n-. Anher calculus resul ha he mahemaicians have prvie us wih is he fac ha if wih n being a cnsan hen f ( ) f n n n. In her wrs, if yu are aking he erivaive, wih respec, f raise a pwer, hen all yu have is cpy he pwer wn u frn an reuce he pwer by. The resul is hus, he value f he riginal pwer imes raise he riginal pwer minus. Remember ha example frm he las secin in which x ( ) a an we g as far as a Well, nw yu knw hw ha las par. s, a a () Okay, here we g again. The erivaive f anyhing wih respec ime is he rae f change f ha anyhing. The quaniy x is psiin, s is he rae f change f psiin. Bu he rae f change f psiin is, by efiniin, he velciy. S is v. Thus equain abve is saying ha v a. Recall ha a was sipulae be a cnsan. Nw if he velciy is a cnsan imes he spwach reaing, hen ha cnsan mus be he cnsan rae f change f he velciy. The rae f change f velciy is, by efiniin, he accelerain f he bjec, s, when x ( ) a wih a being a cnsan, we are ealing wih a siuain in which we have a cnsan value f accelerain equal he cnsan a. 4

5 The Derivaive f a Sum f Terms is he Sum f he Derivaives. This is he isribuive rule fr he erivaive perar. Suppse we have sme funcin x() which can be wrien as he sum f hree her funcins f. The rule abve is jus saying ha x ( ) f ( ) ) h( ) f [ ( ) ) h( ) ] f ( ) ) h( ) f g h Suppse fr example ha x ( ) x v a wih x, v, an a being cnsans. Then, x v a x v a 0 v a v v a a an since, he rae f change f psiin, is jus he velciy v, we have v v a 5