Ni hesabu gani formula zake zilikupa shida?

[Freyzem]

eg, let y=Ne^ax, now find dy/dx.

y=Ne^ax , taking Natural log to both side of the equation (I'm taking natural log because "e" is sorely the base of the Log ).

Lny=LnNe^ax
Lny=axLne, Lne=Log of e to base e which equals 1.

Therefore Lny=ax

Differentiating you get, 1/y dy=adx

dy/dx=ay, substitute y=Ne^ax for y into dy/dx,

dy/dx=aNe^ax.


Huo ndio msingi (principal) wa d/dx(Ne^ax) na nduguze wote wenye e^n.

Boss unavyosema huo ndio msingi wa first principle, naamini pengine first principle huielewi vyema!
Hivi ukiulizwa Differentiate e^x using first principle utatakiwa ufanye kwa namna ulivyofanya?
Kama ni hivyo, then find dy/dx using first principle given that y=lnx.

 

[Mokaze]

Boss unavyosema huo ndio msingi wa first principle, naamini pengine first principle huielewi vyema!
Hivi ukiulizwa Differentiate e^x using first principle utatakiwa ufanye kwa namna ulivyofanya?
Kama ni hivyo, then find dy/dx using first principle given that y=lnx.

Sina maana ya "first principle ya differentiation", maana yangu ni first principle kwa ajili ya e^x na nduguze wa mithili hiyo mfano Ne^bx nk, yaani e raised to a certain power. Hiyo y=lnX sio ya nature hiyo ya e^x.

Yaani; kwanini tumejua kuwa d/dx ya e^x=e^x??!--- tumejua kutokana na "msingi" wake,

ie, let y=e^x

taking ln on both sides you get;

ln y=xlne, lne=1

then differentiating wrt x, 1/ydy=1dx
dy/dx=y=e^x

So, dy/dx=e^x. Huo ndio msingi ninaousema kwa case ya e^x nk,

Sio "differentiation" from the first principle.

 

[Freyzem]

Sina maana ya "first principle ya differentiation", maana yangu ni first principle kwa ajili ya e^x na nduguze wa mithili hiyo mfano Ne^bx nk, yaani e raised to a certain power. Hiyo y=lnX sio ya nature hiyo ya e^x.

Yaani; kwanini tumejua kuwa d/dx ya e^x=e^x??!--- tumejua kutokana na "msingi" wake,

ie, let y=e^x

taking ln on both sides you get;

ln y=xlne, lne=1

then differentiating wrt x, 1/ydy=1dx
dy/dx=y=e^x

So, dy/dx=e^x. Huo ndio msingi ninaousema kwa case ya e^x nk,

Sio "differentiation" from the first principle.

Hujajibu nilichokuuliza!
Bado unazunguka pale pale tu...
Unajichanganya mara unasema sio differentiation fro first principle angali tayari huko juu ushasema d(e^x)/dx=e^x kutokana na first principle ambapo umeonyesha!
Lakini katika kuonyesha kwako hakuna sehemu uliyotumia first principle!
Na ili uniprove wrong, nimekuambia differentiate y=lnx kwa kutumia first principle uliyosema umetumia huko juu!
Kama unafahamu hesabu vizuri, hakuna first principle ya special case kama e^x kama unavyosema wewe!
First principle ndio msingi wa differentiation na inahusisha uwepo wa limits!

 

[Mokaze]

First principle ndio msingi wa differentiation na inahusisha uwepo wa limits!

🤣🤣🤣 by the way what is dy/dx??

dy/dx=∆y/∆x limit as x tends to 0, that means even dy/dx involves limits.

Kama mambo yenyewe ndio hayo, ndio maana sijaona bingwa wa mahesabu (a prominent mathematician) Muafrika, Waalimu wetu wana cram hadi nukta sembuse wanafunzi!!🤣

 

[Teknocrat]

Swali ni hili; hicho kitu hapo juu umekipata wapi???🤣

,

the derivative of exponential function e^x is given by e^x ln e = e^x (because ln e = 1). since the derivative of exponential function e^x is the function itself, then, if f(x) = e^x, then f'(x) = e^x.

hii chini hapa ndio first principle ya differentiation inahusisha limits, Freyzem yuko sawa, you keep complicate yourself na first principle, , japo mimi sio mkali hivyo

 

[Mokaze]

,

the derivative of exponential function e^x is given by e^x ln e = e^x (because ln e = 1). since the derivative of exponential function e^x is the function itself, then, if f(x) = e^x, then f'(x) = e^x.

hii chini hapa ndio first principle ya differentiation inahusisha limits, Freyzem yuko sawa, you keep complicate yourself na first principle, , japo mimi sio mkali hivyo

View attachment 2020272

Mimi sipingi kwamba hiyo the first principle hai involve limits, hiyo principle kwa njia nyingine ndio hii;
eg,given a fuction, y=x
(y+∆y)=(x+∆x) na kuendelea....kufikia dy/dx.

hadi unafika kwenye ∆y/dx limit as x ➡ to 0= dy/dx.

Kwa maana hata hiyo dy/dx=e^x yenyewe ambayo ni final answer ni sawa useme;

∆y/∆x limit as x ➡0=e^x. , sasa hilo jibu mbona lina involve limit, je nalo ni the first principle ??!😀

Juu ya yote;
tuje kwenye mada kwenye hiyo hiyo the first principle of differenciation, let y=e^x, using ln,

taking ln on both sides , ➡lny=x
Put the small increments;
ln(y+∆y)=(x+∆x), na kuendelea----hadi dy/dx

Wewe huoni hiyo ni mtiririko uleule/similar niliouonyesha jinsi ya kutafuta; dy/dy kwa e^x na nduguze pale nilipotumia natural logarithm??! --- kutambua hili haihitaji mtu awe a Math Wallah.---- Maths yahitaji akili na ubunifu na ni art yenye mbinu nyingi, kitu hichi Waafrika wengi hatuna hadi tubadilike ndipo nasi tutawapata akina Terence Tao.

 

[Freyzem]

,

the derivative of exponential function e^x is given by e^x ln e = e^x (because ln e = 1). since the derivative of exponential function e^x is the function itself, then, if f(x) = e^x, then f'(x) = e^x.

hii chini hapa ndio first principle ya differentiation inahusisha limits, Freyzem yuko sawa, you keep complicate yourself na first principle, , japo mimi sio mkali hivyo

View attachment 2020272

Upo sahihi, na hicho ndicho nilikuwa na urgue!
Jamaa anacomplicate issue ambayo ni ndogo sana!

 

[Freyzem]

Mimi sipingi kwamba hiyo the first principle hai involve limits, hiyo principle kwa njia nyingine ndio hii;
eg,given a fuction, y=x
(y+∆y)=(x+∆x) na kuendelea....kufikia dy/dx.

hadi unafika kwenye ∆y/dx limit as x [emoji654] to 0= dy/dx.

Kwa maana hata hiyo dy/dx=e^x yenyewe ambayo ni final answer ni sawa useme;

∆y/∆x limit as x [emoji654]0=e^x. , sasa hilo jibu mbona lina involve limit, je nalo ni the first principle ??![emoji3]

Juu ya yote;
tuje kwenye mada kwenye hiyo hiyo the first principle of differenciation, let y=e^x, using ln,

taking ln on both sides , [emoji654]lny=x
Put the small increments;
ln(y+∆y)=(x+∆x), na kuendelea----hadi dy/dx

Wewe huoni hiyo ni mtiririko uleule/similar niliouonyesha jinsi ya kutafuta; dy/dy kwa e^x na nduguze pale nilipotumia natural logarithm??! --- kutambua hili haihitaji mtu awe a Math Wallah.---- Maths yahitaji akili na ubunifu na ni art yenye mbinu nyingi, kitu hichi Waafrika wengi hatuna hadi tubadilike ndipo nasi tutawapata akina Terence Tao.

Wee jamaa kama ni mwalimu, huwa unawapoteza sana wanafunzi wako! Utakuwa unafanya mambo kwao yawe magumu sana...

 

[Mokaze]

Wee jamaa kama ni mwalimu, huwa unawapoteza sana wanafunzi wako! Utakuwa unafanya mambo kwao yawe magumu sana...

Mimi sio Mwalimu na nimesoma hivi vitu zaidi ya 15yrs ago, ningekuwa Mwalimu ningehakikisha wanafunzi wanafikiri nje ya Box., maths is an art in nature na ndio maana bado ninayokumbukumbu japo sio sana katika Maths.

Sio maths tu hata Physics.

 

[Sex addict]

Number theory

Hii mbona sexy sana mkuu

 

[Sex addict]

Mimi ni combinatorics ilA naikazania tu hivyo hivyo ntachomoa tu

 

[Jeong do jeon]

Schrodinger equation na madude ya quantum physics

 

[Sex addict]

Schrodinger equation na madude ya quantum physics

Ulipiga quantum mechanics mkuuu 🤔

 

[SUKAH]

Mimi sio Mwalimu na nimesoma hivi vitu zaidi ya 15yrs ago, ningekuwa Mwalimu ningehakikisha wanafunzi wanafikiri nje ya Box., maths is an art in nature na ndio maana bado ninayokumbukumbu japo sio sana katika Maths.

Sio maths tu hata Physics.

Embu tueleze ungetumia mbinu gani kwa mfano ili watoto wafikiri nje ya box, pengine kuna walimu humu wakajifunza kutoka kwako na kwenda kupractise huko mashuleni.

 

[Stuxnet]

Kuna na hii:
Coefficient of correlation

 

[Mokaze]

Embu tueleze ungetumia mbinu gani kwa mfano ili watoto wafikiri nje ya box, pengine kuna walimu humu wakajifunza kutoka kwako na kwenda kupractise huko mashuleni.

Nje ya box la kukariri formulas.