Su'aalo Tusaale ah iyo Doodo ku saabsan Waxyaabaha ka soo jeeda Hawlaha Trigonometric
Derivative waa fikrad aasaasi ah oo ku jirta xisaabinta, oo inta badan loo isticmaalo in lagu qeexo heerka isbeddelka shaqada. Marka laga hadlayo hawlaha trigonometric, derivative wuxuu naga caawiyaa inaan fahanno sida isbeddellada xaglaha u saameeyaan qiimaha shaqada. Maqaalkan, waxaan ka wada hadli doonnaa dhowr tusaale oo dhibaatooyin iyo xalal ah oo la xiriira derivatives-ka hawlaha trigonometric.
Hordhac ku saabsan Shaqooyinka Trigonometric
Hawlaha ugu muhiimsan ee trigonometric-ka ee sida caadiga ah loo isticmaalo waxaa ka mid ah sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec), iyo cotangent (cot). Shaqo kastaa waxay leedahay derivative gaar ah:
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. \( \frac{d}{dx} \cot(x) = -\csc^2(x) \)
Fahamkan aasaasiga ah, waxaan u gudbi karnaa dhibaatooyinka iyo xalalka tusaalaha ah ee qoto dheer.
Su'aal Tusaale ah 1: Soo-saarista Shaqada Sine
Soal
Soo hel tarjumaadda shaqada \( f(x) = 3\sin(x) \).
Xalka
Si aan u helno derivative-ka shaqada \( f(x) = 3\sin(x) \), waxaan isticmaali karnaa xeerarka aasaasiga ah ee derivatives-ka iyo sidoo kale joogtada ku jira kalkulus-ka. Derivative-ka \( \sin(x) \) waa \( \cos(x) \).
\[
f'(x) = 3 \cdot \frac{d}{dx} \sin(x) = 3\cos(x)
\]
Markaa, beddelka \( f(x) = 3\sin(x) \) waa \( 3\cos(x) \).
Tusaale 2: Isku-darka Hawlaha Sine iyo Cosine
Soal
Soo hel tarjumaadda shaqada \( g(x) = 2\sin(x) + 4\cos(x) \).
Xalka
Si aan u helno derivative-ka shaqada \( g(x) = 2\sin(x) + 4\cos(x) \), waxaan isticmaali karnaa xeerarka aasaasiga ah ee derivative-ka oo aan aqoonsan karnaa derivative kasta oo ka mid ah \( \sin(x) \) iyo \( \cos(x) \).
\[
g'(x) = 2 \cdot \frac{d}{dx} \sin(x) + 4 \cdot \frac{d}{dx} \cos(x)
\]
Waan ognahay taas:
\[
\frac{d}{dx} \sin(x) = \cos(x)
\]
\[
\frac{d}{dx} \cos(x) = -\sin(x)
\]
Sidaas darteed:
\[
g'(x) = 2 \cos(x) + 4(-\sin(x)) = 2\cos(x) - 4\sin(x)
\]
Markaa, beddelka \( g(x) = 2\sin(x) + 4\cos(x) \) waa \( 2\cos(x) – 4\sin(x) \).
Tusaale 3: Shaqada Labajibbaaran ee Sine
Soal
Soo hel tarjumaadda shaqada \( h(x) = (\sin(x))^2 \).
Xalka
Si loo helo tarjumaadda shaqada \( h(x) = (\sin(x))^2 \), waxaan isticmaali karnaa xeerka silsiladda.
Marka hore, waxaan dejineynaa \( u = \sin(x) \), si \( h(x) = u^2 \).
Waxaan ognahay in asalka \(u^2 \) marka loo eego \(u \) uu yahay \(2u \), asalka \(u \) marka loo eego \(x \) uu yahay \( \cos(x) \).
Markaa,
\[
\frac{d}{dx} (\sin(x))^2 = 2 (\sin(x)) \cdot \cos(x)
\]
Markaa, waxa ka soo jeeda \( h(x) = (\sin(x))^2 \) waa \( 2\sin(x)\cos(x) \).
Su'aal Tusaale ah 4: Shaqada Tagent
Soal
Soo hel tarjumaadda shaqada \( f(x) = \tan(x) \).
Xalka
Si loo helo dheegasho ka timid \( f(x) = \tan(x) \), waxaan isticmaalnaa qeexidda dheegasho ka timid tangent.
\[
\frac{d}{dx} \tan(x) = \sec^2(x)
\]
Markaa, beddelka \( f(x) = \tan(x) \) waa \( \sec^2(x) \).
Tusaale 5: Isku-darka Hawlaha Tangent iyo Secant
Soal
Soo hel tarjumaadda shaqada \( p(x) = \tan(x)\sec(x) \).
Xalka
Si aan u helno wax soo saarka laba functions, waa inaan isticmaalnaa xeerka wax soo saarka.
\[
(fg)' = f'g + fg'
\]
Halkee \( f(x) = \tan(x) \) iyo \( g(x) = \sec(x) \).
Waan ognahay taas:
\[
f'(x) = \sec^2(x)
\]
\[
g'(x) = \sec(x)\tan(x)
\]
Sidaas darteed:
\[
p'(x) = \tan(x) \cdot \sec(x) \tan(x) + \sec(x) \cdot \sec^2(x)
\]
\[
p'(x) = \sec^2(x) \tan^2(x) + \sec^3(x)
\]
Markaa, beddelka \( p(x) = \tan(x)\sec(x) \) waa \( \sec^2(x) \tan^2(x) + \sec^3(x) \).
Su'aal Tusaale ah 6: Falalka Cosecant iyo Cotangent
Soal
Soo hel tarjumaadda shaqada \( q(x) = \csc(x) – \cot(x) \).
Xalka
Si loo helo dheegaanka \( q(x) = \csc(x) – \cot(x) \), waxaan isticmaalnaa qeexitaannada dheegaanka ee cosecant iyo cotangent.
\[
\frac{d}{dx} \csc(x) = -\csc(x) \cot(x)
\]
\[
\frac{d}{dx} \cot(x) = -\csc^2(x)
\]
Sidaas darteed:
\[
q'(x) = -\csc(x)\cot(x) - (-\csc^2(x))
\]
\[
q'(x) = -\csc(x)\cot(x) + \csc^2(x)
\]
Markaa, beddelka \( q(x) = \csc(x) – \cot(x) \) waa \( -\csc(x)\cot(x) + \csc^2(x) \).
Gabagabo
Maqaalkan, waxaan ka wada hadalnay tusaalooyin iyo xalal kala duwan oo la xiriira noocyada kala duwan ee hawlaha trigonometric. Laga soo bilaabo hawlaha aasaasiga ah sida sine iyo cosine, ilaa isku-darka aadka u adag sida badeecada tangent iyo secant, iyo noocyada kala duwan ee cosecant iyo cotangent. Fahmidda noocyada kala duwan ee hawlaha trigonometric ma aha oo kaliya mid faa'iido leh xisaabta saafiga ah laakiin sidoo kale waxay leedahay codsiyo ballaaran oo ku saabsan fiisigiska, injineernimada, iyo qaybaha kale ee kala duwan ee isticmaala isbeddelka shaqada iyo heerarka isbeddelka.
Marka aan ku celcelinno dhibaatooyin badan, fahamkeenna ku saabsan shaqooyinka trigonometric-ka ayaa fiicnaan doona. Waxaan rajeyneynaa in maqaalkani uu kaa caawin doono inaad fahamto fikradda iyo codsiyada derivatives-ka ee shaqooyinka trigonometric-ka!
