Imibuzo eyisibonelo kanye nengxoxo yezakhiwo zemisebenzi esuselwe kokunye
I-derivative yomsebenzi ingumqondo oyisisekelo ekubaleni owusizo kakhulu ekuhlaziyeni ukuziphatha kwemisebenzi ethile. Kulesi sihloko, sizoxoxa ngezinkinga eziningana zezibonelo futhi sixoxe ngezakhiwo ze-derivative yomsebenzi.
Isingeniso kuma-Function Derivatives
I-derivative yomsebenzi \( f \) ivezwa njenge \( f'(x) \). I-derivative yokuqala yomsebenzi inikeza izinga lokushintsha komsebenzi maqondana ne-variable yayo ezimele. Elinye igama elivame ukusetshenziswa lingumehluko. Uma \( y = f(x) \), khona-ke i-derivative ye \( f \) maqondana ne \( x \) ithi:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Izakhiwo Zezinto Ezivela Kumsebenzi
Ezinye izakhiwo ezibalulekile ze-derivative yomsebenzi yilezi:
1. Ukulingana: Uma \( f(x) \) kanye \( g(x) \) kuyimisebenzi ehlukanisekayo, futhi \( c \) kuyinto engaguquki, khona-ke:
\[
\frac{d}{dx} [cf(x) + g(x)] = c f'(x) + g'(x)
\]
2. Umthetho We-Chain: Womsebenzi ohlanganisiwe \( g(f(x)) \):
\[
\frac{d}{dx} g(f(x)) = g'(f(x)) \cdot f'(x)
\]
3. Umkhiqizo: Ngemisebenzi \( u(x) \) kanye \( v(x) \):
\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
4. I-Quotient: Ngemisebenzi \( u(x) \) kanye \( v(x) \) lapho \( v(x) \neq 0 \):
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) – u(x)v'(x)}{(v(x))^2}
\]
Imibuzo Eyisibonelo Nengxoxo
Isibonelo 1: Ukunquma i-Derivative ye-Simple Function
Ake sithi \( f(x) = 3x^2 + 5x – 4 \). Nquma i-derivative yomsebenzi.
Isixazululo:
Sizosebenzisa imithetho eyisisekelo yokuhlukanisa.
\[
f(x) = 3x^2 + 5x – 4
\]
I-derivative yokuqala:
\[
f'(x) = \frac{d}{dx} (3x^2) + \frac{d}{dx} (5x) – \frac{d}{dx} (4)
\]
Ukubala i-derivative ngayinye:
\[
\frac{d}{dx} (3x^2) = 6x
\]
\[
\frac{d}{dx} (5x) = 5
\]
\[
\frac{d}{dx} (4) = 0
\]
Ukuze:
\[
f'(x) = 6x + 5
\]
Isibonelo 2: Ukusebenzisa Umthetho We-Chain
Uma ubheka umsebenzi \( y = (2x^3 – x^2 + 1)^5 \). Thola i-derivative yomsebenzi.
Isixazululo:
Sebenzisa umthetho weketanga. Ake sithi \( u = 2x^3 – x^2 + 1 \), khona-ke umsebenzi ungabhalwa kabusha njengo \( y = u^5 \).
Okokuqala, thola i-derivative ka-\( y \) maqondana no-\( u \):
\[
\frac{dy}{du} = 5u^4
\]
Okulandelayo, thola i-derivative ka-\( u \) maqondana ne-\( x \):
\[
u = 2x^3 – x^2 + 1
\]
\[
\frac{du}{dx} = 6x^2 – 2x
\]
Hlanganisa lezi zinto ezimbili ezisuselwe ku-derivatives nomthetho we-chain:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot (6x^2 – 2x)
\]
Faka futhi esikhundleni \( u = 2x^3 – x^2 + 1 \):
\[
\frac{dy}{dx} = 5(2x^3 – x^2 + 1)^4 \cdot (6x^2 – 2x)
\]
Isibonelo 3: Ukusebenzisa Imithetho Yomkhiqizo
Uma kunikezwe \( f(x) = x^2 e^x \). Nquma i-derivative yomsebenzi.
Isixazululo:
Sebenzisa umthetho womkhiqizo, okungukuthi, uma \( u(x) = x^2 \) kanye \( v(x) = e^x \), bese:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Okokuqala, bala ama-derivatives ka-\( u(x) \) kanye no-\( v(x) \):
\[
u(x) = x^2 \kusho u'(x) = 2x
\]
\[
v(x) = e^x \kusho v'(x) = e^x
\]
Ngokusebenzisa imithetho yomkhiqizo:
\[
f'(x) = 2x \cdot e^x + x^2 \cdot e^x = e^x (2x + x^2)
\]
Isibonelo 4: Ukusebenzisa Umthetho We-Quotient
Kunikezwe \( f(x) = \frac{x^2 + 1}{x + 2} \). Thola i-derivative yomsebenzi.
Isixazululo:
Sebenzisa umthetho we-quotient, okungukuthi uma \( u(x) = x^2 + 1 \) kanye \( v(x) = x + 2 \), bese:
\[
f'(x) = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Okokuqala, bala ama-derivatives ka-\( u(x) \) kanye no-\( v(x) \):
\[
u(x) = x^2 + 1 \kusho u'(x) = 2x
\]
\[
v(x) = x + 2 \kusho v'(x) = 1
\]
Ngokusebenzisa umthetho we-quotient:
\[
f'(x) = \frac{2x(x + 2) – (x^2 + 1)(1)}{(x + 2)^2}
\]
\[
f'(x) = \frac{2x^2 + 4x – x^2 – 1}{(x + 2)^2}
\]
\[
f'(x) = \frac{x^2 + 4x – 1}{(x + 2)^2}
\]
Isiphetho
Ekubaleni, ukuqonda umqondo oyisisekelo wama-derivatives kanye nezakhiwo zawo kubalulekile ekuxazululeni izinkinga ezahlukene zezibalo. Lesi sihloko sifingqa izindlela eziningana zokuthola imisebenzi ngokubonisa ukusetshenziswa kwemithetho eyisisekelo efana nokulingana, amaketanga, imikhiqizo, kanye nama-quotients ngezibonelo eziningana kanye nezingxoxo ezinemininingwane. Ngokuqonda nokusebenzisa njalo ama-derivatives, singaba nekhono elikhulu ekuhlaziyeni izinguquko emisebenzini ezimweni ezahlukene.
