Ihe atụ ajụjụ na mkparịta ụka nke njirimara nke ọrụ ndị sitere n'aka ndị ọzọ
Ihe e si na ya nweta ọrụ bụ echiche dị mkpa na mgbakọ na mwepụ nke bara ezigbo uru maka inyocha omume nke ụfọdụ ọrụ. N'isiokwu a, anyị ga-atụle ọtụtụ nsogbu atụ ma tụlee ihe ndị dị na ya nke ihe e si na ya nweta ọrụ.
Okwu Mmalite nke Ihe Ndị E Ji Mee Ọrụ
A na-egosi ihe e si n'ọrụ \(f \) nweta dịka \(f'(x) \). Ihe mbụ e si n'ọrụ nweta bụ mgbanwe ọrụ ahụ n'ihe gbasara mgbanwe onwe ya. Okwu ọzọ a na-ejikarị eme ihe bụ differential. Ọ bụrụ na \( y = f(x) \), mgbe ahụ ihe e si n'ọrụ \(f \) nweta n'ihe gbasara \( x \) bụ:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h} \]
Njirimara nke ihe ndị e ji arụ ọrụ mee
Ụfọdụ ihe dị mkpa nke ihe e si n'ọrụ nweta bụ:
1. Ahịrịokwu: Ọ bụrụ na \( f(x) \) na \( g(x) \) bụ ọrụ dị iche iche, na \( c \) bụ ihe na-agbanwe agbanwe, mgbe ahụ:
\[
\frac{d}{dx} [cf(x) + g(x)] = c f'(x) + g'(x)
\]
2. Iwu Agbụ: Maka ọrụ mejupụtara \( g(f(x)) \):
\[
\frac{d}{dx} g(f(x)) = g'(f(x)) \cdot f'(x)
\]
3. Ngwaahịa: Maka ọrụ \( u(x) \) na \( v(x) \):
\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
4. Oke: Maka ọrụ \( u(x) \) na \( v(x) \) ebe \( v(x) \neq 0 \):
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) – u(x)v'(x)}{(v(x))^2}
\]
Ajụjụ na Mkparịta ụka Ihe Nlereanya
Ihe atụ nke 1: Ịchọpụta ihe sitere na ọrụ dị mfe
Ka e were ya na \( f(x) = 3x^2 + 5x – 4 \). Chọpụta ihe e si na ya nweta site na ọrụ ahụ.
Ngwọta:
Anyị ga-eji iwu ndị bụ isi nke ọdịiche dị iche iche mee ihe.
\[
f(x) = 3x^2 + 5x – 4
\]
Ihe mbụ e si na ya nweta:
\[
f'(x) = \frac{d}{dx} (3x^2) + \frac{dx} (5x) – \frac{d}{dx} (4)
\]
Ịgbakọ ihe nkesa ọ bụla:
\[
\frac{d}{dx} (3x^2) = 6x
\]
\[
\frac{d}{dx} (5x) = 5
\]
\[
\frac{d}{dx} (4) = 0
\]
Ka ọ were:
\[
f'(x) = 6x + 5
\]
Ihe atụ nke abụọ: Iji Iwu Agbụ
E nyere ọrụ ahụ \( y = (2x^3 – x^2 + 1)^5 \). Chọpụta ihe e si na ya nweta ọrụ ahụ.
Ngwọta:
Jiri iwu agbụ ígwè. Ka e were ya na \( u = 2x^3 – x^2 + 1 \), mgbe ahụ enwere ike idegharị ọrụ ahụ dịka \( y = u^5 \).
Nke mbụ, chọta ihe e si n'aka \(y \) nweta n'ihe gbasara \(u \):
\[
\frac{dy}{du} = 5u^4
\]
Na-esote, chọta ihe e si n'aka \(u \) nweta n'ihe gbasara \( x \):
\[
u = 2x^3 – x^2 + 1
\]
\[
\frac{du}{dx} = 6x^2 – 2x
\]
Jikọta ihe abụọ a na-emepụta na iwu agbụ:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot (6x^2 – 2x)
\]
Tinyegharịa ọzọ \( u = 2x^3 – x^2 + 1 \):
\[
\frac{dy}{dx} = 5(2x^3 – x^2 + 1)^4 \cdot (6x^2 – 2x)
\]
Ihe atụ nke 3: Iji Iwu Ngwaahịa
E nyere \( f(x) = x^2 e^x \). Chọpụta ihe e si na ya nweta ọrụ ahụ.
Ngwọta:
Jiri iwu ngwaahịa ahụ, ya bụ, ọ bụrụ na \( u(x) = x^2 \) na \( v(x) = e^x \), mgbe ahụ:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Nke mbụ, gbakọọ ihe ndị e si n'aka \( u(x) \) na \( v(x) \) nweta:
\[
u(x) = x^2 \pụtara u'(x) = 2x
\]
\[
v(x) = e^x \pụtara v'(x) = e^x
\]
Site n'itinye iwu ngwaahịa n'ọrụ:
\[
f'(x) = 2x \cdot e^x + x^2 \cdot e^x = e^x (2x + x^2)
\]
Ihe atụ nke 4: Iji Iwu Ọnụọgụ
E nyere \( f(x) = \frac{x^2 + 1}{x + 2} \). Chọta ihe e si na ya nweta nke ọrụ ahụ.
Ngwọta:
Jiri iwu nke quotient, ya bụ, ọ bụrụ na \( u(x) = x^2 + 1 \) na \( v(x) = x + 2 \), mgbe ahụ:
\[
f'(x) = \frac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2}
\]
Nke mbụ, gbakọọ ihe ndị e si n'aka \( u(x) \) na \( v(x) \) nweta:
\[
u(x) = x^2 + 1 \pụtara u'(x) = 2x
\]
\[
v(x) = x + 2 \pụtara v'(x) = 1
\]
Site n'itinye iwu nke quotient n'ọrụ:
\[
f'(x) = \frac{2x(x + 2) – (x^2 + 1)(1)}{(x + 2)^2}
\]
\[
f'(x) = \frac{2x^2 + 4x – x^2 – 1}{(x + 2)^2}
\]
\[
f'(x) = \frac{x^2 + 4x – 1}{(x + 2)^2}
\]
Mmechi
N'ịkọwapụta ihe ndị bụ isi nke ihe ndị sitere na ya na ihe ndị ha ji arụ ọrụ dị oke mkpa maka idozi nsogbu mgbakọ na mwepụ dị iche iche. Isiokwu a na-achịkọta ọtụtụ ụzọ maka inweta ọrụ site n'igosi ojiji nke iwu ndị bụ isi dịka linearity, agbụ, ngwaahịa, na quotients site na ọtụtụ ihe atụ na mkparịta ụka zuru ezu. Site n'ịghọta na ime ihe ndị sitere na ya ugboro ugboro, anyị nwere ike ịmụtakwu ihe n'inyocha mgbanwe na ọrụ dị iche iche.
