Mehlala ea Lipotso le Puisano ea Li-derivative tsa Mesebetsi ea Trigonometric
Derivative ke mohopolo oa motheo ho calculus, hangata o sebelisoang ho hlalosa sekhahla sa phetoho ea mosebetsi. Tabeng ea mesebetsi ea trigonometric, derivative e re thusa ho utloisisa hore na liphetoho tsa likhutlo li ama boleng ba mosebetsi joang. Sehloohong sena, re tla tšohla mehlala e 'maloa ea mathata le litharollo tse amanang le derivatives ea mesebetsi ea trigonometric.
Selelekela sa Mesebetsi ea Trigonometric
Mesebetsi e meholo ea trigonometric e sebelisoang hangata e kenyelletsa sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec), le cotangent (cot). Mosebetsi ka mong o na le derivative e itseng:
1. \( \frac{d}{dx} \sin(x) = \cos(x) \)
2. \( \frac{d}{dx} \cos(x) = -\sin(x) \)
3. \( \frac{d}{dx} \tan(x) = \sec^2(x) \)
4. \( \frac{d}{dx} \sec(x) = \sec(x) \tan(x) \)
5. \( \frac{d}{dx} \csc(x) = -\csc(x) \cot(x) \)
6. \( \frac{d}{dx} \cot(x) = -\csc^2(x) \)
Ka kutloisiso ena ea motheo, re ka fetela mathateng a mehlala le litharollo tse tebileng haholoanyane.
Mohlala oa Potso ea 1: Tšimoloho ea Mosebetsi oa Sine
Soal
Fumana derivative ea mosebetsi \( f(x) = 3\sin(x) \).
Penyelesaian
Ho fumana derivative ea mosebetsi \( f(x) = 3\sin(x) \), re ka sebelisa melao ea motheo ea derivatives hammoho le li-constants ka har'a calculus. Derivative ea \( \sin(x) \) ke \( \cos(x) \).
\[
f'(x) = 3 \cdot \frac{d}{dx} \sin(x) = 3\cos(x)
\]
Kahoo, tlhahiso ea \( f(x) = 3\sin(x) \) ke \( 3\cos(x) \).
Mohlala oa 2: Motsoako oa Mesebetsi ea Sine le Cosine
Soal
Fumana derivative ea mosebetsi \( g(x) = 2\sin(x) + 4\cos(x) \).
Penyelesaian
Ho fumana derivative ea mosebetsi \( g(x) = 2\sin(x) + 4\cos(x) \), re ka sebelisa melao ea motheo ea derivative le ho khetholla derivative e 'ngoe le e 'ngoe ea \( \sin(x) \) le \( \cos(x) \).
\[
g'(x) = 2 \cdot \frac{d}{dx} \sin(x) + 4 \cdot \frac{d}{dx} \cos(x)
\]
Rea tseba hore:
\[
\frac{d}{dx} \sin(x) = \cos(x)
\]
\[
\frac{d}{dx} \cos(x) = -\sin(x)
\]
E le hore:
\[
g'(x) = 2 \cos(x) + 4(-\sebe(x)) = 2\cos(x) – 4\sebe(x)
\]
Kahoo, tlhahiso ea \( g(x) = 2\sin(x) + 4\cos(x) \) ke \( 2\cos(x) – 4\sin(x) \).
Mohlala oa 3: Mosebetsi oa Quadratic oa Sine
Soal
Fumana derivative ea mosebetsi \( h(x) = (\sin(x))^2 \).
Penyelesaian
Ho fumana derivative ea mosebetsi \( h(x) = (\sin(x))^2 \), re ka sebelisa molao oa ketane.
Sa pele, re beha \( u = \sin(x) \), e le hore \( h(x) = u^2 \).
Rea tseba hore derivative ea \( u^2 \) mabapi le \( u \) ke \( 2u \), 'me derivative ea \( u \) mabapi le \( x \) ke \( \cos(x) \).
Kahoo,
\[
\frac{d}{dx} (\sin(x))^2 = 2 (\sin(x)) \cdot \cos(x)
\]
Kahoo, tlhahiso ea \( h(x) = (\sin(x))^2 \) ke \( 2\sin(x)\cos(x) \).
Mohlala oa Potso ea 4: Mosebetsi oa Tangent
Soal
Fumana derivative ea mosebetsi \( f(x) = \tan(x) \).
Penyelesaian
Ho fumana derivative ea \( f(x) = \tan(x) \), re sebelisa tlhaloso ea derivative ea tangent.
\[
\frac{d}{dx} \tan(x) = \sec^2(x)
\]
Kahoo, derivative ea \( f(x) = \tan(x) \) ke \( \sec^2(x) \).
Mohlala oa 5: Motsoako oa Mesebetsi ea Tangent le ea Secant
Soal
Fumana derivative ea mosebetsi \( p(x) = \tan(x)\sec(x) \).
Penyelesaian
Ho fumana derivative ea sehlahisoa sa mesebetsi e 'meli, re tlameha ho sebelisa molao oa sehlahisoa.
\[
(fg)' = f'g + fg'
\]
Moo \( f(x) = \tan(x) \) le \( g(x) = \sec(x) \).
Rea tseba hore:
\[
f'(x) = \sec^2(x)
\]
\[
g'(x) = \sec(x)\tan(x)
\]
E le hore:
\[
p'(x) = \tan(x) \cdot \sec(x) \tan(x) + \sec(x) \cdot \sec^2(x)
\]
\[
p'(x) = \sec^2(x) \tan^2(x) + \sec^3(x)
\]
Kahoo, derivative ea \( p(x) = \tan(x)\sec(x) \) ke \( \sec^2(x) \tan^2(x) + \sec^3(x) \).
Mohlala oa Potso ea 6: Mesebetsi ea Cosecant le Cotangent
Soal
Fumana derivative ea mosebetsi \( q(x) = \csc(x) – \cot(x) \).
Penyelesaian
Ho fumana derivative ea \( q(x) = \csc(x) – \cot(x) \), re sebelisa litlhaloso tsa derivative ea cosecant le cotangent.
\[
\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)
\]
\[
\frac{d}{dx} \cot(x) = -\csc^2(x)
\]
E le hore:
\[
q'(x) = -\csc(x)\cot(x) – (-\csc^2(x))
\]
\[
q'(x) = -\csc(x)\cot(x) + \csc^2(x)
\]
Kahoo, motsoako oa \( q(x) = \csc(x) – \cot(x) \) ke \( -\csc(x)\cot(x) + \csc^2(x) \).
Qetello
Sehloohong sena, re buisane ka mehlala le ditharollo tse fapaneng tse amanang le di-derivative tsa mesebetsi ya trigonometric. Ho tloha mesebetsing ya motheo jwalo ka sine le cosine, ho ya ho metswako e rarahaneng haholo jwalo ka sehlahiswa sa tangent le secant, le di-derivative tsa cosecant le cotangent. Ho utlwisisa di-derivative tsa mesebetsi ya trigonometric ha ho thuse feela dipalong tse hlwekileng empa hape ho na le ditshebediso tse pharaletseng fisiks, boenjiniere, le mafapheng a mang a fapaneng a sebedisang phetoho ya tshebetso le sekgahla sa phetoho.
Ka ho itloaetsa mathata a mangata, kutlwisiso ea rona ea li-derivative tsa mesebetsi ea trigonometric e tla ntlafala. Re tšepa hore sengoloa sena se tla u thusa ho utloisisa mohopolo le ts'ebeliso ea li-derivatives mesebetsing ea trigonometric!
