Dokar Sarka a cikin Abubuwan da aka samo
Lissafi yana taka muhimmiyar rawa a fannoni daban-daban na rayuwa, tun daga kimiyya zuwa tattalin arziki. Wani muhimmin batu a cikin lissafi shine ra'ayin derivative. Derivatives suna ba da hanyar fahimtar yadda ayyuka ke canzawa yayin da masu canji ke canzawa. Ɗaya daga cikin mahimman ra'ayoyi a cikin differential calculus shine ƙa'idar sarkar, wanda ke ba da hanyar ƙididdige derivatives na ayyukan mahadi. Wannan labarin zai bincika ƙa'idar sarkar a cikin zurfi, daga ma'anarsa zuwa aikace-aikacensa a aikace a fannoni daban-daban.
Ma'anar Dokar Sarka
Dokar sarka tana ɗaya daga cikin ƙa'idodi na asali a cikin lissafin bambanci wanda ake amfani da shi don ƙididdige asalin abubuwan da suka shafi ayyuka biyu ko fiye. A al'ada, idan muna da ayyuka biyu \( f(x) \) da \( g(x) \), kuma muna son nemo asalin aikin haɗaka \( h(x) = f(g(x))) \), to dokar sarka tana cewa:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
A wata ma'anar, domin ƙididdige ma'aunin \(h(x) \), muna buƙatar ninka ma'aunin \(f \) da aka kimanta a \(g(x) \) ta hanyar ma'aunin \(g \).
Fahimtar Tsarin Mulki na Tsarin Sarka
Domin fahimtar ƙa'idar sarka cikin fahimta, yi tunanin cewa muna tafiya a kan hanya mai lanƙwasa. Saurin da muke ci gaba a kan wannan hanyar (wato, abin da ya samo asali daga matsayinmu dangane da lokaci) ya dogara ne akan abubuwa biyu: saurinmu a alkiblar hanya, da kuma gangaren hanya a wani wuri. Hakazalika, a cikin mahallin ƙa'idar sarka, canje-canje a cikin aikin haɗin gwiwa \( h(x) \) yana faruwa ne ta dalilai biyu: yadda \( f \) ke canzawa dangane da \( g \), da kuma yadda \( g \) ke canzawa dangane da \( x \).
Misali Mai Sauƙi
Bari mu dubi wani misali mai sauƙi inda muke amfani da ƙa'idar sarkar don ƙididdige abin da aka samo daga aikin mahaɗi.
A ce \( f(x) = \sin(x) \) da \( g(x) = x^2 \). Sannan, muna son nemo abin da ya samo asali daga \( h(x) = \sin(x^2) \).
Ga matakan:
1. Gano aikin waje \( f \) da aikin ciki \( g \):
– Aikin waje: \( f(u) = \sin(u) \), inda \(u = g(x) \).
– Aikin ciki: \( g(x) = x^2 \).
2. Nemo abubuwan da suka samo asali daga \( f \) da \( g \):
– \( f'(u) = \cos(u) \).
– \( g'(x) = 2x \).
3. Aiwatar da ƙa'idar sarkar:
– \( h'(x) = f'(g(x)) \cdot g'(x) \).
– Don haka, \( h'(x) = \cos(x^2) \cdot 2x \).
Saboda haka, muna da \( h'(x) = 2x \cos(x^2) \).
Aiwatar da Dokar Sarka
Ilimin kimiyyar lissafi
A fannin kimiyyar lissafi, ana amfani da ƙa'idar sarka sau da yawa don ƙididdige gudu da hanzari a cikin tsarin motsi. Misali, idan aka ba da matsayin abu a matsayin aikin lokaci \( s(t) \), kuma wannan matsayin ya dogara da wani canji kamar zafin jiki ko matsin lamba, za mu iya amfani da ƙa'idar sarka don tantance alaƙar da ke tsakanin gudu ko hanzarin abu da wancan canjin.
tattalin arzikin
A fannin tattalin arziki, amfani da ƙa'idar sarkar abu ɗaya ne a cikin nazarin gefe. A wannan yanayin, ribar kamfani ko farashinsa na iya dogara ne akan wasu masu canji da yawa, kamar farashin samfura, farashin samarwa, ko adadin da aka sayar. Ƙa'idar sarkar tana ba mu damar fahimtar yadda canje-canje a cikin kowane ɗayan waɗannan masu canji ke shafar ribar gaba ɗaya ko farashi.
Bambancin Kai Tsaye
Dokar sarka tana da matuƙar muhimmanci a cikin bambance-bambancen da ba a fayyace ba, inda muke mu'amala da ayyukan da ba a fayyace su a sarari ba. A ce muna da lissafin \( x^2 + y^2 = 1 \) wanda ke wakiltar da'irar radius 1. Za mu iya amfani da ƙa'idar sarka don nemo tushen da ba a fayyace ba na \( y \) dangane da \( x \).
Bari mu ɗauki ma'anar ɓangarorin biyu na lissafin:
\[ \frac{d}{dx} (x^2 + y^2) = \frac{d}{dx} (1) \]
Ta amfani da ƙa'idar sarkar, mun sami:
\[ 2x + 2y \frac{dy}{dx} = 0 \]
Magance matsalar \( \frac{dy}{dx} \):
\[ 2y \frac{dy}{dx} = -2x \]
\[ \frac{dy}{dx} = -\frac{x}{y} \]
Wannan misali ne na yadda ƙa'idar sarkar ke samar da kayan aiki mai ƙarfi don bambancewa a ɓoye a cikin shari'o'in da \( y \) aikin \( x \) ne wanda ba a fayyace shi a sarari ba.
Misalai Masu Rikici
A ce \( f(x) = e^{3x^2+2x} \). Muna son nemo wanda aka samo daga \( f(x) \). A wannan yanayin, aikin ciki shine \( u(x) = 3x^2 + 2x \) kuma aikin waje shine \( f(u) = e^u \).
Amfani da ƙa'idar sarkar:
1. Ma'anar aikin waje: \( f'(u) = e^u \).
2. An samo asali daga aikin da ke cikin: \( u'(x) = 6x + 2 \).
Don haka, bisa ga ƙa'idar sarkar:
\[ f'(x) = e^{3x^2+2x} \cdot (6x+2) \]
Kammalawa
Dokar sarka kayan aiki ne mai mahimmanci a cikin lissafin bambanci. Ta hanyar samar da hanyar ƙididdige abin da aka samo daga aikin mahaɗi, dokar sarka tana faɗaɗa fa'idar amfani da abubuwan da aka samo daga ...
A wajen koyo da amfani da ƙa'idar sarka, mabuɗin nasara shine fahimtar tsarin ayyukan da ke tattare da su da kuma takamaiman matakan da ake buƙata don aiwatar da su. Tare da fahimta mai ƙarfi da aiki mai daidaito, ƙa'idar sarka na iya zama kayan aiki mai ƙarfi wajen magance matsaloli daban-daban na lissafi da aikace-aikacen gaske.
