Ukubhala i-Derivative ye-Function
I-Pendahuluan
Kumathematika, ikakhulukazi i-calculus, i-derivative ingumqondo oyisisekelo odlala indima ebalulekile ezinhlobonhlobo zezicelo. Ama-derivatives awasetshenziswa kuphela kumathematika e-theory kodwa nakwesayensi, ubunjiniyela, ezomnotho, nakwezinye izifundo eziningi. Lesi sihloko sizoxoxa ngokuningiliziwe nge-derivative yomsebenzi, sihlanganisa izisekelo zawo, imithetho ebalulekile, kanye nezibonelo zokusetshenziswa.
Izisekelo Zezinto Ezivela Kuwo
Incazelo yama-Derivatives
I-derivative yomsebenzi ichaza izinga lokushintsha komsebenzi maqondana ne-variable yayo ezimele. Ngokwemvelo, i-derivative ingachazwa njengokuthambeka komugqa we-tangent othinta igrafu yomsebenzi endaweni ethile.
Uma \( y = f(x) \), khona-ke i-derivative yokuqala ye-\( f \) maqondana ne-\( x \) iboniswa yi-\( f'(x) \) noma \( \frac{dy}{dx} \). Incazelo esemthethweni ye-derivative inikezwa umkhawulo olandelayo:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Isaziso Esivela Kuwo
Kunezinhlobo eziningana zezimpawu ezivame ukusetshenziswa ekubhaleni amagama ahlobene:
1. Incazelo kaLeibniz: \( \frac{dy}{dx} \)
2. Umbhalo we-Lagrange: \( f'(x) \)
3. Umbhalo kaNewton: \( y' \)
4. Umbhalo we-Euler: \( Df(x) \)
Incazelo ngayinye inokusetshenziswa okuthile kanye nezimo lapho isetshenziswa khona kakhulu.
Imithetho Eyisisekelo Yokwehlukanisa
Imithetho Yokwengeza Nokususa
Uma \( f(x) \) kanye \( g(x) \) kuyimisebenzi emibili ehlukanisekayo, khona-ke:
\[ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \]
Imithetho Yokuphindaphinda
Kwemisebenzi emibili \( u(x) \) kanye \( v(x) \):
\[ \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
Imithetho Yokuhlukanisa
Uma \( u(x) \) kanye \( v(x) \) kuyimisebenzi emibili, kanye \( v(x) \neq 0 \):
\[ \frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x) \cdot v(x) – u(x) \cdot v'(x)}{[v(x)]^2} \]
Umthetho Weketanga
Ngokwakheka kwemisebenzi emibili \( f(u) \) kanye \( u(g) \):
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]
Izibonelo Zokusetshenziswa
Izakhi Zemisebenzi Ye-Polynomial
Ake sithi \( f(x) = 3x^3 – 5x^2 + 2x – 1 \). Ukuze sithole i-derivative yalo msebenzi, sisebenzisa imithetho eyisisekelo yokuhlukanisa.
\[ f'(x) = \frac{d}{dx} (3x^3) – \frac{d}{dx} (5x^2) + \frac{d}{dx} (2x) – \frac{d}{dx} (1) \]
\[ f'(x) = 9x^2 – 10x + 2 \]
Izakhi Zomsebenzi Wokuchaza kanye Nowokuhlela
Uma \( f(x) = e^x \), khona-ke i-derivative yomsebenzi we-exponential ithi:
\[ f'(x) = e^x \]
Ngomsebenzi we-logarithm wemvelo \( f(x) = \ln(x) \):
\[ f'(x) = \frac{1}{x} \]
Izinhlobo zemisebenzi ye-Trigonometric
Ngemisebenzi eyisisekelo ye-trigonometric:
– Uma \( f(x) = \sin(x) \), khona-ke \( f'(x) = \cos(x) \)
– Uma \( f(x) = \cos(x) \), khona-ke \( f'(x) = -\sin(x) \)
– Uma \( f(x) = \tan(x) \), khona-ke \( f'(x) = \sec^2(x) \)
Okususelwa ku-Composite Function
Ake sithi \( f(x) = \sin(2x) \). Singasebenzisa umthetho weketanga:
\[ f'(x) = \cos(2x) \cdot \frac{d}{dx}(2x) = \cos(2x) \cdot 2 = 2\cos(2x) \]
Izithako Ezithuthukisiwe
I-Derivatives Yesibili Nelandelayo
I-derivative yesibili iyi-derivative yomsebenzi wokuqala we-derivative. Uma \( y = f(x) \) khona-ke i-derivative yesibili iboniswa yi-\( f”(x) \) noma \( \frac{d^2y}{dx^2} \). Njalonjalo nge-derivative yesithathu \( f”'(x) \) noma \( \frac{d^3y}{dx^3} \).
Ake sithi \( f(x) = x^4 \):
\[ f'(x) = 4x^3 \]
\[ f”(x) = \frac{d}{dx}(4x^3) = 12x^2 \]
\[ f”'(x) = \frac{d}{dx}(12x^2) = 24x \]
\[ f””(x) = \frac{d}{dx}(24x) = 24 \]
Ukusetshenziswa Kwezinto Ezivela Ku-Physics
Ku-physics, ama-derivatives avame ukusetshenziswa ukunquma ijubane kanye nokusheshisa. Ake sithi i-\( s(t) \) iwumsebenzi wesikhundla maqondana nesikhathi \(t \). Ijubane \( v(t) \) liwumphumela wokuqala wesikhundla:
\[ v(t) = s'(t) \]
Ukusheshisa \( a(t) \) yi-derivative yokuqala yejubane noma i-derivative yesibili yesikhundla:
\[ a(t) = v'(t) = s”(t) \]
Isiphetho
I-derivative yomsebenzi ingumqondo oyisisekelo ekubaleni ngezinhlelo zokusebenza ezibanzi emikhakheni eyahlukene. Ukuqonda okunembile kwe-derivative njengokwehlela komugqa oqondile kunikeza ukuqonda okubalulekile ngezakhiwo kanye nokuziphatha komsebenzi. Ukuqonda nokukwazi ukusebenzisa imithetho yokuhlukanisa njenge-chain rule, umthetho womkhiqizo, kanye nomthetho wokuhlukanisa kubalulekile kunoma ubani ofunda i-calculus. Ngezibonelo ezilula kanye nezinhlelo zokusebenza ku-physics, lesi sihloko sithemba ukunikeza ukuqonda okuphelele kokubhala i-derivative yomsebenzi.
