Su'aalo Tusaale ah oo ka hadlaya Shaqada Qorista Waxyaabaha laga soo qaatay
Kala-soocidda shaqada waa fikrad aasaasi ah oo ku jirta xisaabta, oo inta badan loo isticmaalo qaybaha kala duwan ee sayniska, sida fiisigiska, dhaqaalaha, bayoolajiga, iyo injineernimada. Kala-soocidda shaqada waxay cabbirtaa sida ugu dhakhsaha badan ee qiimaheedu isu beddelo marka loo eego isbeddellada doorsoomayaasha madaxbannaan. Maqaalkan, waxaan kaga hadli doonnaa dhowr dhibaato oo tusaale ah oo ku lug leh qorista kala-soocidda shaqada, oo dhammaystiran sharraxaad.
Su'aal Tusaale ah 1: Soo-saaridda Hawlaha Fudud
Su'aal: Soo hel tarjumaadda koowaad ee shaqada \( f(x) = 3x^2 + 5x + 7 \).
Dood:
Si loo go'aamiyo beddelka ugu horreeya ee shaqada \( f(x) \), waxaan isticmaalnaa xeerarka aasaasiga ah ee kala-soocidda, kuwaas oo kala ah:
\[
\frac{d}{dx}(ax^n) = anx^{n-1}
\]
Markaa, waxaan u xisaabin karnaa ereyga ka soo jeeda erey kasta oo ku jira shaqadan sidan soo socota:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(7)
\]
\[
f'(x) = 3 \cdot 2x^{2-1} + 5 \cdot 1x^{1-1} + 0
\]
\[
f'(x) = 6x + 5
\]
Markaa, beddelka ugu horreeya ee shaqada \( f(x) = 3x^2 + 5x + 7 \) waa \( f'(x) = 6x + 5 \).
Su'aal Tusaale 2: Waxyaabaha laga soo qaatay shaqooyinka Trigonometric
Su'aal: Soo hel tarjumaadda ugu horreysa ee shaqada \( g(x) = \sin(x) + \cos(x) \).
Dood:
Waxaan u isticmaalnaa xeerarka aasaasiga ah ee tarjumaadda ee shaqooyinka trigonometric:
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
\[
\frac{d}{dx}(\cos(x)) = -\sin(x)
\]
Markaa:
\[
g'(x) = \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(\cos(x))
\]
\[
g'(x) = \cos(x) – \sin(x)
\]
Markaa, beddelka ugu horreeya ee shaqada \( g(x) = \sin(x) + \cos(x) \) waa \( g'(x) = \cos(x) – \sin(x) \).
Su'aal Tusaale ah 3: Soo-saarista Shaqada Isku-dhufashada
Su'aal: Soo hel tarjumaadda ugu horreysa ee shaqada \( h(x) = x^2 \sin(x) \).
Dood:
Functions-ka oo ah natiijada laba function, waxaan isticmaalnaa xeerka isku dhufashada:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
Ka soo qaad \( u(x) = x^2 \) iyo \( v(x) = \sin(x) \). Kadib:
\[
u'(x) = \frac{d}{dx}(x^2) = 2x
\]
\[
v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)
\]
Annagoo adeegsanayna xeerka isku dhufashada, waxaan qori karnaa:
\[
h'(x) = [x^2]' \sin(x) + x^2 [\sin(x)]'
\]
\[
h'(x) = 2x \sin(x) + x^2 \cos(x)
\]
Markaa, beddelka ugu horreeya ee shaqada \( h(x) = x^2 \sin(x) \) waa \( h'(x) = 2x \sin(x) + x^2 \cos(x) \).
Su'aal Tusaale ah 4: Soo-saaridda Shaqada Halabuurka
Su'aal: Soo hel tarjumaadda ugu horreysa ee shaqada \( k(x) = \sin(x^2) \).
Dood:
Hawlaha ka kooban laba shaqdood, waxaan isticmaalnaa xeerka silsiladda:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
U ogolow \( f(u) = \sin(u) \) iyo \( u = x^2 \). Kadibna \( f'(u) = \cos(u) \) iyo \( g'(x) = \frac{d}{dx}(x^2) = 2x \).
Annagoo adeegsanayna xeerka silsiladda, waxaan qori karnaa:
\[
k'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x
\]
Markaa, beddelka ugu horreeya ee shaqada \( k(x) = \sin(x^2) \) waa \( k'(x) = 2x \cos(x^2) \).
Tusaale Su'aal 5aad: Kala soocidda Hawlaha Caqliga leh
Dhibaato: Soo hel tarjumaadda ugu horreysa ee shaqada \( m(x) = \frac{2x}{x^2 + 1} \).
Dood:
Functions-ka laba functional ah, waxaan isticmaalnaa xeerka quotient-ka:
\[
\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Ka soo qaad \( u(x) = 2x \) iyo \( v(x) = x^2 + 1 \). Kadib:
\[
u(x) = 2
\]
\[
v'(x) = \frac{d}{dx}(x^2 + 1) = 2x
\]
Annagoo adeegsanayna xeerka saamiga, waxaan qori karnaa:
\[
m'(x) = \frac{[2x]'(x^2 + 1) – 2x[x^2 + 1]'}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2(x^2 + 1) – 2x \cdot 2x}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2x^2 + 2 – 4x^2}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{-(2x^2 – 2)}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2}
\]
Markaa, beddelka ugu horreeya ee shaqada \( m(x) = \frac{2x}{x^2 + 1} \) waa \( m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2} \).
Gabagabo
Maqaalkan, waxaan ka wada hadalnay dhowr tusaale oo ku saabsan dhibaatooyinka ku lug leh noocyada kala duwan ee hawlaha, oo u dhexeeya hawlaha fudud, hawlaha trigonometric, isku dhufashada, isku-darka, iyo hawlaha macquulka ah. Tusaale kasta wuxuu muujinayaa isticmaalka habboon ee xeerarka kala duwan, sida xeerka aasaasiga ah, xeerka silsiladda, xeerka isku dhufashada, iyo xeerka saamiga. Fahmidda sida loo dabaqo xeerarkan ayaa muhiim u ah xallinta dhibaatooyinka kala duwan ee xisaabinta. Ku celcelinta iyo tababarka ayaa kaa caawin doona xoojinta fahamkaaga iyo xirfadahaaga kala soocida hawlaha.
