Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o nā Derivatives o nā hana Trigonometric
He manaʻo nui ka derivative i ka calculus, i hoʻohana pinepine ʻia e wehewehe i ka wikiwiki o ka loli o kahi hana. I ke ʻano o nā hana trigonometric, kōkua ka derivative iā mākou e hoʻomaopopo pehea e hoʻopilikia ai nā loli i nā kihi i ka waiwai o ka hana. Ma kēia ʻatikala, e kūkākūkā mākou i kekahi mau pilikia hoʻohālike a me nā hoʻonā e pili ana i nā derivatives o nā hana trigonometric.
Hoʻolauna i nā Hana Trigonometric
ʻO nā hana trigonometric nui i hoʻohana pinepine ʻia e komo pū ana me ka sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec), a me cotangent (cot). Loaʻa i kēlā me kēia hana kahi derivative kikoʻī:
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. \( \frac{d}{dx} \cot(x) = -\csc^2(x) \)
Me kēia ʻike maʻalahi, hiki iā mākou ke neʻe aku i nā pilikia a me nā hoʻonā hohonu aʻe.
Laʻana Nīnau 1: Ka hopena o ka Hana Sine
Nīnau
E huli i ka derivative o ka hana \( f(x) = 3\sin(x) \).
Hoʻoponopono
No ka loaʻa ʻana o ka derivative o ka hana \( f(x) = 3\sin(x) \), hiki iā mākou ke hoʻohana i nā lula kumu o nā derivatives a me nā constants i ka calculus. ʻO ka derivative o \( \sin(x) \) ʻo \( \cos(x) \).
\[
f'(x) = 3 \cdot \frac{d}{dx} \sin(x) = 3\cos(x)
\]
No laila, ʻo ka derivative o \( f(x) = 3\sin(x) \) ʻo \( 3\cos(x) \).
Laʻana 2: Hui pū ʻana o nā Hana Sine a me Cosine
Nīnau
E huli i ka derivative o ka hana \( g(x) = 2\sin(x) + 4\cos(x) \).
Hoʻoponopono
No ka loaʻa ʻana o ka derivative o ka hana \( g(x) = 2\sin(x) + 4\cos(x) \), hiki iā mākou ke hoʻohana i nā lula derivative kumu a ʻike i kēlā me kēia derivative o \( \sin(x) \) a me \( \cos(x) \).
\[
g'(x) = 2 \cdot \frac{d}{dx} \sin(x) + 4 \cdot \frac{d}{dx} \cos(x)
\]
Ua ʻike mākou:
\[
\frac{d}{dx} \sin(x) = \cos(x)
\]
\[
\frac{d}{dx} \cos(x) = -\sin(x)
\]
No laila:
\[
g'(x) = 2 \cos(x) + 4(-\sin(x)) = 2\cos(x) – 4\sin(x)
\]
No laila, ʻo ka derivative o \( g(x) = 2\sin(x) + 4\cos(x) \) ʻo \( 2\cos(x) – 4\sin(x) \).
Laʻana 3: Hana Quadratic o Sine
Nīnau
E huli i ka derivative o ka hana \( h(x) = (\sin(x))^2 \).
Hoʻoponopono
No ka loaʻa ʻana o ka derivative o ka hana \( h(x) = (\sin(x))^2 \), hiki iā kākou ke hoʻohana i ke kānāwai kaulahao.
ʻO ka mea mua, hoʻonohonoho mākou iā \( u = \sin(x) \), i hiki ai iā \( h(x) = u^2 \).
Ua ʻike mākou ʻo ka derivative o \( u^2 \) e pili ana iā \( u \) ʻo ia hoʻi \( 2u \), a ʻo ka derivative o \( u \) e pili ana iā \( x \) ʻo ia hoʻi \( \cos(x) \).
No laila,
\[
\frac{d}{dx} (\sin(x))^2 = 2 (\sin(x)) \cdot \cos(x)
\]
No laila, ʻo ka derivative o \( h(x) = (\sin(x))^2 \) ʻo \( 2\sin(x)\cos(x) \).
Laʻana Nīnau 4: Hana Tangent
Nīnau
E huli i ka derivative o ka hana \( f(x) = \tan(x) \).
Hoʻoponopono
No ka loaʻa ʻana o ka derivative o \( f(x) = \tan(x) \), hoʻohana mākou i ka wehewehena o ka derivative o ka tangent.
\[
\frac{d}{dx} \tan(x) = \sec^2(x)
\]
No laila, ʻo ka derivative o \( f(x) = \tan(x) \) ʻo \( \sec^2(x) \).
Laʻana 5: Hoʻohuihui ʻia o nā Hana Tangent a me Secant
Nīnau
E huli i ka derivative o ka hana \( p(x) = \tan(x)\sec(x) \).
Hoʻoponopono
No ka loaʻa ʻana o ka derivative o ka huahana o nā hana ʻelua, pono mākou e hoʻohana i ka lula huahana.
\[
(fg)' = f'g + fg'
\]
Ma kahi o \( f(x) = \tan(x) \) a me \( g(x) = \sec(x) \).
Ua ʻike mākou:
\[
f'(x) = \sec^2(x)
\]
\[
g'(x) = \sec(x)\tan(x)
\]
No laila:
\[
p'(x) = \tan(x) \cdot \sec(x) \tan(x) + \sec(x) \cdot \sec^2(x)
\]
\[
p'(x) = \sec^2(x) \tan^2(x) + \sec^3(x)
\]
No laila, ʻo ka derivative o \( p(x) = \tan(x)\sec(x) \) ʻo ia \( \sec^2(x) \tan^2(x) + \sec^3(x) \).
Laʻana Nīnau 6: Nā Hana Cosecant a me Cotangent
Nīnau
E huli i ka derivative o ka hana \( q(x) = \csc(x) – \cot(x) \).
Hoʻoponopono
No ka loaʻa ʻana o ka derivative o \( q(x) = \csc(x) – \cot(x) \), hoʻohana mākou i nā wehewehena o ka derivative o ka cosecant a me ka cotangent.
\[
\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)
\]
\[
\frac{d}{dx} \cot(x) = -\csc^2(x)
\]
No laila:
\[
q'(x) = -\csc(x)\cot(x) – (-\csc^2(x))
\]
\[
q'(x) = -\csc(x)\cot(x) + \csc^2(x)
\]
No laila, ʻo ka derivative o \( q(x) = \csc(x) – \cot(x) \) ʻo ia \( -\csc(x)\cot(x) + \csc^2(x) \).
Ka hopena
Ma kēia ʻatikala, ua kūkākūkā mākou i nā ʻano hoʻohālike like ʻole a me nā hopena e pili ana i nā derivatives o nā hana trigonometric. Mai nā hana kumu e like me ka sine a me ka cosine, a hiki i nā hui paʻakikī e like me ka huahana o ka tangent a me ka secant, a me nā derivatives o cosecant a me cotangent. ʻO ka hoʻomaopopo ʻana i nā derivatives o nā hana trigonometric ʻaʻole wale ia he mea pono i ka makemakika maʻemaʻe akā he mau noi ākea hoʻi i ka physics, ka ʻenekinia, a me nā ʻano ʻē aʻe like ʻole e hoʻohana ana i ka loli hana a me nā wikiwiki o ka loli.
Ma ka hoʻomaʻamaʻa ʻana i nā pilikia hou aku, e hoʻomaikaʻi ʻia ko mākou ʻike ʻana i nā derivatives o nā hana trigonometric. Manaʻolana, e kōkua kēia ʻatikala iā ʻoe e hoʻomaopopo i ke kumumanaʻo a me nā noi o nā derivatives i nā hana trigonometric!
