Tambayoyi Misali Game da Ayyukan Rubutu Abubuwan da aka samo
Asalin aikin wani muhimmin abu ne a cikin lissafi, wanda ake amfani da shi akai-akai a fannoni daban-daban na kimiyya, kamar kimiyyar lissafi, tattalin arziki, ilmin halitta, da injiniyanci. Asalin aikin yana auna yadda ƙimarsa ke canzawa da sauri dangane da canje-canje a cikin masu canjinsa masu zaman kansu. A cikin wannan labarin, za mu tattauna misalai da yawa na matsaloli da suka shafi rubuta asalin aikin, tare da bayani.
Misali Tambaya ta 1: An samo daga Ayyuka Masu Sauƙi
Tambaya: Nemo farkon abin da aka samo daga aikin \( f(x) = 3x^2 + 5x + 7 \).
Tattaunawa:
Domin tantance asalin farko na aikin \( f(x) \), muna amfani da ƙa'idodin asali na bambance-bambance, wato:
\[
\frac{d}{dx}(ax^n) = anx^{n-1}
\]
Don haka, za mu iya ƙididdige tushen kowane kalma a cikin aikin kamar haka:
\[
f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) + \frac{d}{dx}(7)
\]
\[
f'(x) = 3 \cdot 2x^{2-1} + 5 \cdot 1x^{1-1} + 0
\]
\[
f'(x) = 6x + 5
\]
Don haka, farkon abin da aka samo daga aikin \( f(x) = 3x^2 + 5x + 7 \) shine \( f'(x) = 6x + 5 \).
Misali Tambaya ta 2: Abubuwan da aka samo daga Ayyukan Trigonometric
Tambaya: Nemo farkon abin da aka samo daga aikin \( g(x) = \sin(x) + \cos(x) \).
Tattaunawa:
Muna amfani da ƙa'idodin asali na asali don ayyukan trigonometric:
\[
\frac{d}{dx}(\sin(x)) = \cos(x)
\]
\[
\frac{d}{dx}(\cos(x)) = -\sin(x)
\]
Don haka:
\[
g'(x) = \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(\cos(x))
\]
\[
g'(x) = \cos(x) – \sin(x)
\]
Don haka, farkon abin da aka samo daga aikin \( g(x) = \sin(x) + \cos(x) \) shine \( g'(x) = \cos(x) – \sin(x) \).
Misali Tambaya ta 3: Tsarin Aikin ninkawa
Tambaya: Nemo farkon abin da aka samo daga aikin \( h(x) = x^2 \sin(x) \).
Tattaunawa:
Ga ayyuka waɗanda suka samo asali daga ayyuka biyu, muna amfani da ƙa'idar ninkawa:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
A ce \( u(x) = x^2 \) da kuma \( v(x) = \sin(x) \). Sannan:
\[
u'(x) = \frac{d}{dx}(x^2) = 2x
\]
\[
v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)
\]
Ta amfani da ƙa'idar ninkawa, za mu iya rubuta:
\[
h'(x) = [x^2]' \sin(x) + x^2 [\sin(x)]'
\]
\[
h'(x) = 2x \sin(x) + x^2 \cos(x)
\]
Don haka, farkon abin da aka samo daga aikin \( h(x) = x^2 \sin(x) \) shine \( h'(x) = 2x \sin(x) + x^2 \cos(x) \).
Misali Tambaya ta 4: Tsarin Aikin Haɗawa
Tambaya: Nemo farkon abin da aka samo daga aikin \( k(x) = \sin(x^2) \).
Tattaunawa:
Ga ayyukan da suka ƙunshi ayyuka biyu, muna amfani da ƙa'idar sarkar:
\[
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]
Bari \(f(u) = \sin(u) \) da \(u = x^2 \). Sannan \(f'(u) = \cos(u) \) da kuma \( g'(x) = \frac{d}{dx}(x^2) = 2x \).
Ta amfani da ƙa'idar sarkar, za mu iya rubuta:
\[
k'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x
\]
Don haka, farkon abin da aka samo daga aikin \( k(x) = \sin(x^2) \) shine \( k'(x) = 2x \cos(x^2) \).
Misali Tambaya ta 5: An samo daga Ayyukan Hankali
Matsala: Nemo farkon abin da aka samo daga aikin \( m(x) = \frac{2x}{x^2 + 1} \).
Tattaunawa:
Ga ayyuka waɗanda sune ƙimar ayyuka guda biyu, muna amfani da ƙa'idar ƙimar ayyuka:
\[
\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
A ce \( u(x) = 2x \) da kuma \( v(x) = x^2 + 1 \). Sannan:
\[
u(x) = 2
\]
\[
v'(x) = \frac{d}{dx}(x^2 + 1) = 2x
\]
Ta amfani da ƙa'idar quotient, za mu iya rubutawa:
\[
m'(x) = \frac{[2x]'(x^2 + 1) – 2x[x^2 + 1]'}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2(x^2 + 1) – 2x \cdot 2x}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2x^2 + 2 – 4x^2}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{-(2x^2 – 2)}{(x^2 + 1)^2}
\]
\[
m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2}
\]
Don haka, farkon abin da aka samo daga aikin \( m(x) = \frac{2x}{x^2 + 1} \) shine \( m'(x) = \frac{2 – 2x^2}{(x^2 + 1)^2} \).
Kammalawa
A cikin wannan labarin, mun tattauna misalai da dama na matsalolin da suka shafi abubuwan da suka samo asali daga ayyuka, tun daga ayyuka masu sauƙi, ayyukan trigonometric, ninkawa, tsari, da ayyukan hankali. Kowane misali yana nuna yadda ake amfani da ƙa'idodin da suka samo asali, kamar ƙa'idar asali, ƙa'idar sarka, ƙa'idar ninkawa, da ƙa'idar quotient. Fahimtar yadda ake amfani da waɗannan ƙa'idodi yana da mahimmanci don magance matsalolin lissafi masu rikitarwa a fannoni daban-daban. Maimaita aiki da horo zai taimaka wajen ƙarfafa fahimtarka da ƙwarewarka wajen bambance ayyuka.
