Mibvunzo yeMienzaniso neKukurukurirana kweZvinobva muMabasa eTrigonometric
Chinobva pane chimwe chinhu (derivative) ipfungwa huru mukuverenga, inowanzoshandiswa kutsanangura mwero wekuchinja kwebasa. Panyaya yemabasa etrigonometric, chinobva pane chimwe chinhu (derivative) chinotibatsira kunzwisisa kuti shanduko mumakona dzinokanganisa sei kukosha kwebasa. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yematambudziko nemhinduro dzine chekuita nemaderivative emabasa etrigonometric.
Nhanganyaya kuMabasa eTrigonometric
Mashandiro makuru etrigonometric anowanzo shandiswa anosanganisira sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec), uye cotangent (cot). Basa rega rega rine derivative chaiyo:
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. \( \frac{d}{dx} \cot(x) = -\csc^2(x) \)
Nekunzwisisa uku kwekutanga, tinogona kuenderera mberi kune mamwe matambudziko emuenzaniso nemhinduro dzakadzama.
Muenzaniso Mubvunzo 1: Kubva paSine Function
Mubvunzo
Tsvaga derivative yebasa \( f(x) = 3\sin(x) \).
Penyelesaian
Kuti tiwane derivative yebasa \( f(x) = 3\sin(x) \), tinogona kushandisa mitemo yekutanga yema derivatives pamwe chete nema constants mu calculus. Derivative ye \( \sin(x) \) ndi \( \cos(x) \).
\[
f'(x) = 3 \cdot \frac{d}{dx} \sin(x) = 3\cos(x)
\]
Saka, chinobva pa \( f(x) = 3\sin(x) \) ndi \( 3\cos(x) \).
Muenzaniso 2: Musanganiswa weSine neCosine Functions
Mubvunzo
Tsvaga derivative yebasa \( g(x) = 2\sin(x) + 4\cos(x) \).
Penyelesaian
Kuti tiwane derivative yebasa \( g(x) = 2\sin(x) + 4\cos(x) \), tinogona kushandisa mitemo yekutanga derivative uye kuziva derivative yega yega ye \( \sin(x) \) uye \( \cos(x) \).
\[
g'(x) = 2 \cdot \frac{d}{dx} \sin(x) + 4 \cdot \frac{d}{dx} \cos(x)
\]
Tinoziva kuti:
\[
\frac{d}{dx} \sin(x) = \cos(x)
\]
\[
\frac{d}{dx} \cos(x) = -\sin(x)
\]
Kuti:
\[
g'(x) = 2 \cos(x) + 4(-\chivi(x)) = 2\cos(x) – 4\chivi(x)
\]
Saka, chinobva pa \( g(x) = 2\sin(x) + 4\cos(x) \) ndi \( 2\cos(x) – 4\sin(x) \).
Muenzaniso 3: Quadratic Function yeSine
Mubvunzo
Tsvaga derivative yebasa \( h(x) = (\sin(x))^2 \).
Penyelesaian
Kuti tiwane derivative yebasa \( h(x) = (\sin(x))^2 \), tinogona kushandisa chain rule.
Kutanga, tinoisa \( u = \sin(x) \), kuitira kuti \( h(x) = u^2 \).
Tinoziva kuti chinobva pa \( u^2 \) maererano ne \( u \) ndi \( 2u \), uye chinobva pa \( u \) maererano ne \( x \) ndi \( \cos(x) \).
Saka,
\[
\frac{d}{dx} (\sin(x))^2 = 2 (\sin(x)) \cdot \cos(x)
\]
Saka, chinobva pa \( h(x) = (\sin(x))^2 \) ndi \( 2\sin(x)\cos(x) \).
Muenzaniso Mubvunzo 4: Basa reTangent
Mubvunzo
Tsvaga derivative yebasa \( f(x) = \tan(x) \).
Penyelesaian
Kuti tiwane derivative ye \( f(x) = \tan(x) \), tinoshandisa tsananguro ye derivative ye tangent.
\[
\frac{d}{dx} \tan(x) = \sec^2(x)
\]
Saka, chinobva pa \( f(x) = \tan(x) \) ndi \( \sec^2(x) \).
Muenzaniso 5: Musanganiswa weMabasa eTangent neSecant
Mubvunzo
Tsvaga derivative yebasa \( p(x) = \tan(x)\sec(x) \).
Penyelesaian
Kuti tiwane chinobva pachinhu chiri mumabasa maviri, tinofanira kushandisa mutemo wechigadzirwa.
\[
(fg)' = f'g + fg'
\]
Kupi \( f(x) = \tan(x) \) uye \( g(x) = \sec(x) \).
Tinoziva kuti:
\[
f'(x) = \sec^2(x)
\]
\[
g'(x) = \sec(x)\tan(x)
\]
Kuti:
\[
p'(x) = \tan(x) \cdot \sec(x) \tan(x) + \sec(x) \cdot \sec^2(x)
\]
\[
p'(x) = \sec^2(x) \tan^2(x) + \sec^3(x)
\]
Saka, chinobva pa \( p(x) = \tan(x)\sec(x) \) ndi \( \sec^2(x) \tan^2(x) + \sec^3(x) \).
Muenzaniso Mubvunzo 6: Mabasa eCosecant neCotangent
Mubvunzo
Tsvaga derivative yebasa \( q(x) = \csc(x) – \cot(x) \).
Penyelesaian
Kuti tiwane derivative ye \( q(x) = \csc(x) – \cot(x) \), tinoshandisa tsananguro ye derivative yecosecant necotangent.
\[
\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)
\]
\[
\frac{d}{dx} \cot(x) = -\csc^2(x)
\]
Kuti:
\[
q'(x) = -\csc(x)\cot(x) – (-\csc^2(x))
\]
\[
q'(x) = -\csc(x)\cot(x) + \csc^2(x)
\]
Saka, chinobva pa \( q(x) = \csc(x) – \cot(x) \) ndi \( -\csc(x)\cot(x) + \csc^2(x) \).
Mhedziso
Muchinyorwa chino, takurukura mienzaniso yakasiyana-siyana nemhinduro dzine chekuita nemabviro emabasa etrigonometric. Kubva kumabasa ekutanga akadai sesine necosine, kusvika kusanganiswa kwakaomarara senge chigadzirwa chetangent nesecant, uye mabviro ecosecant necotangent. Kunzwisisa mabviro emabasa etrigonometric hakungobatsiri chete mumasvomhu chete asiwo kune mashandisirwo akakura mufizikisi, mainjiniya, nedzimwe nzvimbo dzakasiyana-siyana dzinoshandisa shanduko yekushanda uye mwero wekuchinja.
Nekudzidzira matambudziko akawanda, kunzwisisa kwedu ma derivatives emabasa e trigonometric kuchawedzera. Tinovimba kuti chinyorwa chino chichakubatsira kunzwisisa pfungwa uye mashandisirwo ema derivatives mumabasa e trigonometric!
