Imibuzo Eyisibonelo Exoxa Ngezincazelo Zomsebenzi Wokubhala
I-derivative yomsebenzi ingumqondo oyisisekelo ekubaleni, ovame ukusetshenziswa emikhakheni ehlukahlukene yesayensi, njenge-physics, ezomnotho, i-biology, kanye nobunjiniyela. I-derivative yomsebenzi ilinganisa ukuthi inani layo lishintsha ngokushesha kangakanani maqondana nezinguquko eziguquguqukayo zayo ezizimele. Kulesi sihloko, sizoxoxa ngezibonelo eziningana zezinkinga ezihilela ukubhala i-derivative yomsebenzi, ephelele nezincazelo.
Isibonelo Umbuzo 1: Okususelwe Kumisebenzi Elula
Umbuzo: Thola i-derivative yokuqala yomsebenzi \( f(x) = 3x^2 + 5x + 7 \).
Ingxoxo:
Ukuze sithole i-derivative yokuqala yomsebenzi \( f(x) \), sisebenzisa imithetho eyisisekelo yokuhlukanisa, okungukuthi:
\[
\frac{d}{dx}(ax^n) = anx^{n-1}
\]
Ngakho-ke, singabala i-derivative yethemu ngayinye kumsebenzi kanje:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(7)
\]
\[
f'(x) = 3 \cdot 2x^{2-1} + 5 \cdot 1x^{1-1} + 0
\]
\[
f'(x) = 6x + 5
\]
Ngakho-ke, i-derivative yokuqala yomsebenzi \( f(x) = 3x^2 + 5x + 7 \) ngu \( f'(x) = 6x + 5 \).
Isibonelo Umbuzo 2: Izakhi Zomsebenzi We-Trigonometric
Umbuzo: Thola i-derivative yokuqala yomsebenzi \( g(x) = \sin(x) + \cos(x) \).
Ingxoxo:
Sisebenzisa imithetho eyisisekelo yokususelwa emisebenzini ye-trigonometric:
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
\[
\frac{d}{dx}(\cos(x)) = -\sin(x)
\]
Ngakho-ke:
\[
g'(x) = \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(\cos(x))
\]
\[
g'(x) = \cos(x) – \sin(x)
\]
Ngakho-ke, i-derivative yokuqala yomsebenzi \( g(x) = \sin(x) + \cos(x) \) ngu \( g'(x) = \cos(x) – \sin(x) \).
Isibonelo Umbuzo 3: Umsebenzi Wokuphindaphinda Osuselwe Kuwo
Umbuzo: Thola i-derivative yokuqala yomsebenzi \( h(x) = x^2 \sin(x) \).
Ingxoxo:
Kuma-function angumkhiqizo wemisebenzi emibili, sisebenzisa umthetho wokuphindaphinda:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
Ake sithi \( u(x) = x^2 \) kanye \( v(x) = \sin(x) \). Bese:
\[
u'(x) = \frac{d}{dx}(x^2) = 2x
\]
\[
v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)
\]
Sisebenzisa umthetho wokuphindaphinda, singabhala:
\[
h'(x) = [x^2]' \sin(x) + x^2 [\sin(x)]'
\]
\[
h'(x) = 2x \sin(x) + x^2 \cos(x)
\]
Ngakho-ke, i-derivative yokuqala yomsebenzi \( h(x) = x^2 \sin(x) \) ngu \( h'(x) = 2x \sin(x) + x^2 \cos(x) \).
Isibonelo Umbuzo 4: Incazelo Yomsebenzi Wokuqamba
Umbuzo: Thola i-derivative yokuqala yomsebenzi \( k(x) = \sin(x^2) \).
Ingxoxo:
Kumisebenzi ehlanganisa imisebenzi emibili, sisebenzisa umthetho weketanga:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
Ake \( f(u) = \sin(u) \) kanye \( u = x^2 \). Bese \( f'(u) = \cos(u) \) kanye \( g'(x) = \frac{d}{dx}(x^2) = 2x \).
Sisebenzisa umthetho weketanga, singabhala:
\[
k'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x
\]
Ngakho-ke, i-derivative yokuqala yomsebenzi \( k(x) = \sin(x^2) \) ngu \( k'(x) = 2x \cos(x^2) \).
Isibonelo Umbuzo 5: Ukwehlukaniswa Kwemisebenzi Enengqondo
Inkinga: Thola i-derivative yokuqala yomsebenzi \( m(x) = \frac{2x}{x^2 + 1} \).
Ingxoxo:
Kuma-functions angu-quotient wemisebenzi emibili, sisebenzisa umthetho we-quotient:
\[
\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Ake sithi \( u(x) = 2x \) kanye \( v(x) = x^2 + 1 \). Bese:
\[
u'(x) = 2
\]
\[
v'(x) = \frac{d}{dx}(x^2 + 1) = 2x
\]
Sisebenzisa umthetho we-quotient, singabhala:
\[
m'(x) = \frac{[2x]'(x^2 + 1) – 2x[x^2 + 1]'}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2(x^2 + 1) – 2x \cdot 2x}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2x^2 + 2 – 4x^2}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{-(2x^2 – 2)}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2}
\]
Ngakho-ke, i-derivative yokuqala yomsebenzi \( m(x) = \frac{2x}{x^2 + 1} \) ngu \( m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2} \).
Isiphetho
Kulesi sihloko, sixoxe ngezibonelo eziningana zezinkinga ezihilela ama-derivatives emisebenzi, kusukela kwimisebenzi elula, imisebenzi ye-trigonometric, ukuphindaphinda, ukwakheka, kanye nemisebenzi enengqondo. Isibonelo ngasinye sibonisa ukusetshenziswa okufanele kwemithetho ye-derivative, njengomthetho oyisisekelo, umthetho we-chain, umthetho wokuphindaphinda, kanye nomthetho we-quotient. Ukuqonda ukuthi ungayisebenzisa kanjani le mithetho kubalulekile ekuxazululeni izinkinga zokubala eziyinkimbinkimbi kakhulu kuzo zonke izigaba ezahlukene. Ukuzijwayeza nokuqeqeshwa okuphindaphindiwe kuzosiza ukuqinisa ukuqonda kwakho namakhono okuhlukanisa imisebenzi.
