Lanjutan Materi (5) Turunan fungsi Trigonometri (Matematika Peminatan Kelas XII)

 $\text{D. Aturan Rantai Turunan Fungsi Trigonometri}$

Jika fungsi  $y=(f\circ g)(x)=f(g(x))=f(u)$  dengan  $u=g(x)$, maka turunan dari fungsi komposisi tersebut adalah:

$\begin{array}{c}{y}^{\prime }={(f\circ g)}^{\prime }(x)=f^{\prime }(g(x))\times g^{\prime }(x)\\ \\ \mathbf{\text{atau}}\\ \\ {\displaystyle \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}}\end{array}$

Perluasan dari teorema di atas, adalah berikut:

Diberikan $y=(f\circ g\circ h)(x)=h\left(f(g(x))\right)=f(u)$  dengan  $u=g(v)$  dan  $v=h(x)$, maka turunan pertama dari fungsi komposisi tersebut adalah:

$\begin{array}{c}{y}^{\prime }={(f\circ g\circ h)}^{\prime }(x)=f^{\prime }\left(g(h(x))\right)\times g^{\prime }(h(x))\times h^{\prime }(x)\\ \\ \mathbf{\text{atau}}\\ \\ {\displaystyle \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dv}\times \frac{dv}{dx}}\end{array}$

$\boxed{\text{CONTOH SOAL}}$

$\begin{array}{l}\\ 1.& \text{Tentukan turunan pertama dari}\\ & f(x)={\sin }^{20}(8x^5+\pi )\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}f(x)& ={\sin }^{20}(8x^5+\pi )={(\sin (8x^5+\pi ))}^{20}\\ \text{Dim}& \text{isalkan}\\ y& =u^{20},\text{dengan}\\ u& =\sin (8x^5+\pi )\text{serta}u=\sin v\\ & \text{dan}v=(8x^5+\pi ),\\ \text{mak}& \text{a}\\ {\displaystyle \frac{dy}{du}}& =20u^{19}=20{\sin }^{19}(8x^5+\pi ),\\ {\displaystyle \frac{du}{dv}}& =\cos v=\cos (8x^5+\pi ),\\ {\displaystyle \frac{dv}{dx}}& =40x^4\\ \text{Seh}& \text{ingga}\\ f^{\prime }(x)& ={\displaystyle \frac{dy}{dx}}\\ & ={\displaystyle \frac{dy}{du}\times \frac{du}{dv}\times \frac{dv}{dx}}\\ & =20{\sin }^{19}(8x^5+\pi )\times \cos (8x^5+\pi )\times 40x^4\\ & =800x^4{\sin }^{19}(8x^5+\pi )\cos (8x^5+\pi )\end{array}\\ \\ & \mathbf{\text{atau kalau ingin langsungan saja}}\\ & \text{Tentunya jika Anda sudah lancar adalah}\\ \\ & \begin{array}{rl}f(x)& ={\sin }^{20}(8x^5+\pi )\\ f^{\prime }(x)& =20({\sin }^{19}(8x^5+\pi ))\times \cos (8x^5+\pi )\times \left(40x^4\right)\\ & =800x^4{\sin }^{19}(8x^5+\pi )\cos (8x^5+\pi )\end{array}\end{array}$

$\begin{array}{l}\\ 2.& \text{Tentukan turunan pertama dari}\\ & g(x)=\sqrt[5]{{\cos }^3(x^2-\pi )}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}g(x)& =\sqrt[5]{{\cos }^3(x^2-\pi )}={\cos }^{{.}^{\frac{3}{5}}}(x^2-\pi )\\ \text{Dim}& \text{isalkan}\\ y& =u^{{.}^{\frac{3}{5}}},\text{dengan}\\ u& =\cos (x^2-\pi )\text{serta}u=\cos v\\ & \text{dan}v=(x^2-\pi ),\\ \text{mak}& \text{a}\\ {\displaystyle \frac{dy}{du}}& ={\displaystyle \frac{3}{5}{u}^{{.}^{-\frac{2}{5}}}=\frac{3}{5}{\cos }^{{.}^{-\frac{2}{5}}}(x^2-\pi ),}\\ {\displaystyle \frac{du}{dv}}& =-\sin v=-\sin (x^2-\pi ),\\ {\displaystyle \frac{dv}{dx}}& =2x\\ \text{Seh}& \text{ingga}\\ g^{\prime }(x)& ={\displaystyle \frac{dy}{dx}}\\ & ={\displaystyle \frac{dy}{du}\times \frac{du}{dv}\times \frac{dv}{dx}}\\ & ={\displaystyle \frac{3}{5}{\cos }^{{.}^{-\frac{2}{5}}}(x^2-\pi )\times (-\sin (x^2-\pi ))\times (2x)}\\ & =-{\displaystyle \frac{6x\sin (x^2-\pi )}{5{\cos }^{{.}^{\frac{2}{5}}}(x^2-\pi )}}\\ & =-{\displaystyle \frac{6x\sin (x^2-\pi )}{5\sqrt[5]{{\cos }^2(x^2-\pi )}}}\end{array}\\ \\ & \mathbf{\text{atau kalau ingin langsungan saja}}\\ \\ & \begin{array}{rl}g(x)& =\sqrt[5]{{\cos }^3(x^2-\pi )}={\cos }^{{.}^{\frac{3}{5}}}(x^2-\pi )\\ g^{\prime }(x)& ={\displaystyle \frac{3}{5}{\cos }^{{.}^{-\frac{2}{5}}}(x^2-\pi )\times (-\sin (x^2-\pi ))\times (2x)}\\ & =-{\displaystyle \frac{6x\sin (x^2-\pi )}{5{\cos }^{{.}^{\frac{2}{5}}}(x^2-\pi )}}\\ & =-{\displaystyle \frac{6x\sin (x^2-\pi )}{5\sqrt[5]{{\cos }^2(x^2-\pi )}}}\end{array}\end{array}$

$\begin{array}{l}\\ 3.& \text{Tentukan turunan pertama dari}\\ & h(x)=\cos (\sin x^{2020})\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}h(x)& =\cos (\sin x^{2020})\\ \text{Dim}& \text{isalkan}\\ y& =\cos (\sin x^{2020})=\cos u,\text{dengan}\\ u& =\sin x^{2020}=\sin v,\text{serta}v=x^{2020}\\ \text{mak}& \text{a}\\ {\displaystyle \frac{dy}{du}}& =-\sin u=-\sin (\sin x^{2020}),\\ & \text{atau}dy=-\sin udu\\ {\displaystyle \frac{du}{dv}}& =\cos v\text{atau}du=\cos vdv\\ {\displaystyle \frac{dv}{dx}}& =2020x^{2019}\text{atau}dv=2020x^{2019}dx\\ \text{Seh}& \text{ingga}\\ h^{\prime }(x)& ={\displaystyle \frac{dy}{dx}}\\ & ={\displaystyle \frac{dy}{du}\times \frac{du}{dv}\times \frac{dv}{dx}}\\ & =-\sin (\sin x^{2020})\times \cos x^{2020}\times \left(2020x^{2019}\right)\\ & =-2020x^{2019}\sin (\sin x^{2020})\cos x^{2020}\end{array}\\ \\ & \mathbf{\text{atau}}\\ & \begin{array}{rl}dy& =-\sin udu\\ & =-\sin u\times \cos vdv\\ & =-\sin u\times \cos v\times \left(2020x^{2019}\right)dx\\ {\displaystyle \frac{dy}{dx}}& =-\sin u\times \cos v\times \left(2020x^{2019}\right)\\ & =.......(\text{tinggal dimasukkan})\end{array}\end{array}$

DAFTAR PUSTAKA

1. Wirodikromo, S. 2007. Matematika Jilid 2 IPA untuk Kelas XI Berdasarkan Standar Isi 2006. Jakarta: ERLANGGA.