Li-derivative tsa Mesebetsi ea Trigonometric
Lipalong tse tsoetseng pele, haholo-holo calculus, hangata re kopana le mesebetsi ea trigonometric e kang sine (sin), cosine (cos), secant (sec), cosecant (csc), tangent (tan), le cotangent (cot). Moelelong ona, ho tseba metsoako ea mesebetsi ena ho bohlokoa, haholo-holo bakeng sa lits'ebetso tsa fisiks, boenjiniere le saense ea khomphutha. Sengoloa sena se tla hlalosa ka botlalo mokhoa oa ho fumana metsoako ea mesebetsi ena ea trigonometric.
Selelekela ho Derivatives
Pele re buisana ka di-derivative tsa mesebetsi ya trigonometric, ha re hlahlobeng ka bokgutshwane mohopolo wa derivative. Derivative ya mosebetsi e re fa sekgahla sa phetoho ya mosebetsi oo mabapi le phetoho ya wona e ikemetseng. Ka mantswe a geometri, derivative ya mosebetsi f(x) ntlheng ya x e fana ka gradient, kapa leralla, la mola o tangent ho ya ho curve f(x) ntlheng eo.
Ho ya ka dipalo, derivative ya pele ya mosebetsi f(x) e hlaloswa ka tsela e latelang:
\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) – f(x)}{\Delta x} \]
Tlhaloso ena e ntse e tšoana bakeng sa mesebetsi ea trigonometric, empa ho tla ba bonolo haeba re tseba li-derivatives tse ling tsa motheo tsa mesebetsi ea motheo ea trigonometric.
Li-derivative tsa Mesebetsi ea Motheo ea Trigonometric
1. Setho sa sine (sin x)
Mosebetsi oa sine ke o mong oa mesebetsi ea motheo ea trigonometric. Tšimoloho ea sin x ke cos x. Sena se nkiloe meeling e itseng le algebra e fapaneng.
\[ \frac{d}{dx}(\sin x) = \cos x \]
Ke hore, haeba f(x) = sin x, joale f'(x) = cos x.
2. Letswai la Cosine (cos x)
Cosine ke mosebetsi o mong oa motheo oa trigonometric. Tšimoloho ea cos x ke -sin x.
\[ \frac{d}{dx}(\cos x) = -\sin x \]
Ke hore, haeba f(x) = cos x, joale f'(x) = -sin x.
3. Letswai le nang le sebopeho sa Tangent (lesela le letšo x)
Mosebetsi wa tangent ke karolelano ya sine le cosine. Derivative ya tan x ke sec^2 x. Sena se ka fumanwa ho sebediswa molao wa derivative bakeng sa mesebetsi ya composite (ketane).
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]
Ke hore, haeba f(x) = tan x, joale f'(x) = sec² x.
4. Cotangent Derivative (kamore e nyane x)
Cotangent ke ntho e fapaneng le tangent. Karolo e tswang ho cot x ke -csc² x.
\[ \frac{d}{dx}(\cot x) = -\csc^2 x \]
Ke hore, haeba f(x) = cot x, joale f'(x) = -csc² x.
5. Secant Derivative (sec x)
Mosebetsi oa secant o fapane le cosine. Setho sa sec x ke sec x tan x.
\[ \frac{d}{dx}(\sec x) = \sec x \tan x \]
Ke hore, haeba f(x) = sec x, joale f'(x) = sec x tan x.
6. Kosecant Derivative (csc x)
Mosebetsi oa cosecant ke o fapaneng le sine. Setho sa csc x ke -csc x cot x.
\[ \frac{d}{dx}(\csc x) = -\csc x \cot x \]
Ke hore, haeba f(x) = csc x, joale f'(x) = -csc x cot x.
Tšebeliso ea Melao e Tsoang ho Mesebetsi ea Trigonometric
Hang ha re se re tseba di-derivative tsa motheo tsa mesebetsi ya trigonometric, re ka atoloha ho ya ditshebedisong tse rarahaneng haholoanyane re sebedisa melao ya di-derivative jwalo ka molao wa ketane, molao wa sehlahiswa, le molao wa kakaretso.
1. Molao oa Ketane
Molao oa ketane o sebelisoa ha re e-na le mosebetsi o nang le mesebetsi e 'meli kapa ho feta. Mehlala ea tšebeliso ea oona:
Haeba re na le mosebetsi \( g(x) = \sin(3x^2) \), re ka sebedisa molao wa ketane ho fumana derivative ya wona:
\[ g'(x) = \frac{d}{dx}[\sin(3x^2)] \]
\[ = \cos(3x^2) \cdot \frac{d}{dx}[3x^2] \]
\[ = \cos(3x^2) \cdot 6x \]
\[ = 6x \cos(3x^2) \]
2. Melao ea Sehlahisoa
Molao oa sehlahisoa o sebelisoa ha re e-na le mosebetsi o leng sehlahisoa sa mesebetsi e 'meli kapa ho feta. Mehlala ea tšebeliso ea oona:
Haeba \( h(x) = x^2 \sin(x) \), ho ya ka molao wa sehlahiswa:
\[ h'(x) = \frac{d}{dx}[x^2 \cdot \sin(x)] \]
\[ = x^2 \cdot \cos(x) + \sin(x) \cdot \frac{d}{dx}[x^2] \]
\[ = x^2 \cos(x) + \sin(x) \cdot 2x \]
\[ = x^2 \cos(x) + 2x \sin(x) \]
3. Melao ea Lipalo
Molao oa kakaretso o sebelisoa ha re e-na le mosebetsi o leng kakaretso ea mesebetsi e 'meli kapa ho feta. Mehlala ea tšebeliso ea oona:
Haeba \( f(x) = \sin(x) + \cos(x) \):
\[ f'(x) = \frac{d}{dx}[\sin(x) + \cos(x)] \]
\[ = \frac{d}{dx}[\sin(x)] + \frac{d}{dx}[\cos(x)] \]
\[ = \cos(x) + (-\sin(x)) \]
\[ = \cos(x) – \sin(x) \]
Mesebetsi ea Trigonometric e Fetohileng le Li-derivative tsa Tsona
Ntle le mesebetsi ea motheo ea trigonometric, re boetse re na le mesebetsi ea trigonometric e fapaneng joalo ka sin^-1 x (arcsin x), cos^-1 x (arccos x), le tan^-1 x (arctan x). Li-derivative tsa mesebetsi ena le tsona li bohlokoa ts'ebelisong ea calculus.
Mohlala:
– Tšimoloho ea arcsin x:
\[ \frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 – x^2}} \]
– Tšimoloho ea arccos x:
\[ \frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 – x^2}} \]
– Tšimoloho ea arctan x:
\[ \frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2} \]
Qetello
Ho ithuta di-derivative tsa mesebetsi ya trigonometric ke kgato ya motheo ho calculus. Di-derivative tsa mesebetsi ya motheo jwalo ka sin, cos, tan, cot, sec, le csc di fana ka motheo o tiileng wa ho sekaseka le ho rarolla mathata a rarahaneng haholo mafapheng a fapaneng. Ho feta moo, kutlwisiso ya molao wa ketane, molao wa sehlahiswa, le molao wa kakaretso e re thusa ho sebetsana le di-derivative tsa mesebetsi e rarahaneng haholo. Tsebo ena e bohlokwa haholo ditshebedisong tse ngata tse sebetsang le tsa thuto, ho kenyeletswa fisiks, boenjiniere le saense ya dikhomphutha.
