d

56dc514e-4886-49ce-bef7-36cf44e85c39 44 4 5 false false 0 decimal

&[DATE] &[TIME] - &[FILENAME]

d

Derivative utility X\032E\ t\032E\ n\032E\ p\032E\ DIFF R\0324\ 0 : R\0324\ 0 : k\0324\ 0 : 13 mat x\0324\ X\032E\ str\0324\ diff(x\0324\ num2str, t\032E\ num2str, n\032E\ num2str) concat : R\0324\ k\0324\ k\0324\ 1 + : el str\0324\ str2num p\032E\ * : 21 line for k\0324\ 1 ≡ R\0324\ 1 el k\0324\ 0 : r\0324\ 1 X\032E\ rows range c\0324\ 1 X\032E\ cols range A\0324\ r\0324\ c\0324\ el R\0324\ k\0324\ k\0324\ 1 + : el : for for A\0324\ 31 line if 31 line :

d.Aux2 try to catch the differential variable and it's order. t\032E\ d.Aux2 U\0324\ t\032E\ Unknowns : R\0324\ 0 : 12 mat k\0324\ 1 : k\0324\ U\0324\ length ≤ k\0324\ k\0324\ 1 + : str U\0324\ k\0324\ el num2str : Unknowns(dSymbol str) concat str2num num2str 0 ≡ 0 l\0324\ 05 range t\032E\ U\0324\ k\0324\ el l\0324\ 1 + 1 DIFF num2str 0 ≡ break continue if for n\0324\ str dSymbol findstr : (1 - ≡ (R\0324\ k\0324\ 1 el R\0324\ k\0324\ 2 el 12 mat U\0324\ k\0324\ el l\0324\ 12 mat : m\0324\ str 1 n\0324\ rows substr dSymbol findstr : (1 - ≡ R\0324\ k\0324\ 1 el R\0324\ k\0324\ 2 el 12 mat U\0324\ k\0324\ el l\0324\ n\0324\ rows + 1 - 12 mat : R\0324\ k\0324\ 1 el R\0324\ k\0324\ 2 el 12 mat U\0324\ k\0324\ el l\0324\ m\0324\ rows + 1 - 12 mat : if if 21 line if 21 line for R\0324\ 31 line :

d(x,t) returns dx/dt x\032E\ t\032E\ d R\0324\ t\032E\ d.Aux2 : D\0324\ x\032E\ : 12 mat R\0324\ num2str 0 ≡ D\0324\ 0 : k\0324\ 1 R\0324\ rows range D\0324\ D\0324\ R\0324\ k\0324\ 1 el R\0324\ k\0324\ 2 el 1 DIFF : for if D\0324\ 31 line :

d(x) returns the total differential of x x\032E\ d U\0324\ x\032E\ Unknowns : s\0324\ 0 : 12 mat k\0324\ 1 : k\0324\ U\0324\ length ≤ k\0324\ k\0324\ 1 + : R\0324\ U\0324\ k\0324\ el d.Aux2 : R\0324\ num2str 0 ≡ 0 du\0324\ dn\0324\ 12 mat dSymbol U\0324\ k\0324\ el num2str concat str2num d.Aux2 : du\0324\ du\0324\ num2str num2str dSymbol dSymbol concat dSymbol strrep str2num : s\0324\ s\0324\ x\032E\ R\0324\ 1 el R\0324\ 2 el du\0324\ dn\0324\^DIFF + : 31 line if 21 line for s\0324\ 31 line :

t C P D d R sys(: k1 1 C length range R R dSymbol C k1 el num2str :0, concat : for k2 1 P length range R R P k2 el num2str': dSymbol P k2 el num2str / dSymbol t num2str, concat : for k3 1 D length range R R dSymbol D k3 el num2str : D k3 el num2str'* dSymbol t num2str, concat : for R 1, k1 k2 + k3 + 1 + num2str,1) concat 51 line :

Differentials

dSymbol d :

Enter an expression, and say that have some constants

z x y 2^* y a x * / + :

da 0 :

dz z d : y 1 x 2^y * a * - (* dx * - 12 x 2^* y * a * + (dy * x * + a x 2^* /

Total differential

Partial derivatives

dz dx d y 1 x 2^y * a * - (* a x 2^* / -

dz dy d 12 x 2^* y * a * + x a * /

z x d y 1 x 2^y * a * - (* a x 2^* / -

z y d 12 x 2^* y * a * + a x * /

Notice that

dz dy 2^d 0

because

dz dy d dy d 0

but

dz y 2^d 2 dx *

Some examples

dSymbol \2202\. :

r θ cos * r θ sin * 21 mat d θ cos ∂.r * r θ sin * ∂.θ * - θ sin ∂.r * r θ cos * ∂.θ * + 21 mat

dSymbol \2202\ :

∂a 0 :

a e b x *^* d e b x *^a * x ∂b * b ∂x * + (*

a e x sin^* d e x sin^x cos * a * ∂x *

x 2^x y * ln * d x 12 x y * ln * + (∂x * y * x ∂y * + (* y /

∂a Clear 1

d1 a b * d : b ∂a * a ∂b * +

d2 d1 d : ∂a ∂b b ∂a * + ∂b + (* a ∂b 2^* +

d3 d2 d : 2 ∂b b ∂a * + (∂a 2^* ∂a ∂b a * + (∂b 2^* + (* ∂b ∂a * ∂b ∂a + (* +

∂a 0 :

d1 a b * d : a ∂b *

d2 d1 d : a ∂b 2^*

d3 d2 d : 2 ∂b 3^* a *

Chain rule

c a x g f * d : a x g f x diff * ∂x *

x f x ln :

x g x cos :

c a x tan * ∂x * -

a x cos ln * d x sin a * ∂x * x cos / -

Total differential

t x x' f d t x x' f t diff ∂t * t x x' f x diff ∂x * + t x x' f x' diff ∂x' * +

Alvaro