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

 $\color{blue}\textrm{D. Aturan Rantai Turunan Fungsi Trigonometri}$

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

$\color{blue}\begin{matrix} y'=\left ( f\circ g \right )'(x)=f'\left ( g(x) \right )\times g'(x)\\\\ \color{black}\textbf{atau}\\\\ \displaystyle \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx} \end{matrix}$

Perluasan dari teorema di atas, adalah berikut:

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

$\color{blue}\begin{matrix} y'=\left ( f\circ g\circ h \right )'(x)=f'\left ( g\left ( h(x) \right ) \right )\times g'\left ( h(x) \right )\times h'(x)\\\\ \color{black}\textbf{atau}\\\\ \displaystyle \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dv}\times \frac{dv}{dx} \end{matrix}$

$\LARGE\color{magenta}\fbox{CONTOH SOAL}$

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

$\begin{array}{ll}\\ 2.&\textrm{Tentukan turunan pertama dari}\\ &g(x)=\sqrt[5]{\cos ^{3}\left ( x^{2}-\pi \right )}\\\\ &\textrm{Jawab}:\\ &\color{blue}\begin{aligned}g(x)&=\sqrt[5]{\cos ^{3}\left ( x^{2}-\pi \right )}=\cos ^{.^{\frac{3}{5}}}\left ( x^{2}-\pi \right )\\ \textrm{Dim}&\textrm{isalkan}\\ y& =u^{.^{\frac{3}{5}}},\: \: \textrm{dengan}\\ u&=\cos \left ( x^{2}-\pi \right )\: \: \textrm{serta}\: \: u=\cos v\\ &\textrm{dan}\: \: v=\left ( x^{2}-\pi \right ),\\ \color{black}\textrm{mak}&\color{black}\textrm{a}\\ \displaystyle \frac{dy}{du}&=\displaystyle \frac{3}{5}u^{.^{-\frac{2}{5}}}=\frac{3}{5}\cos ^{.^{-\frac{2}{5}}}\left ( x^{2}-\pi \right ),\\ \displaystyle \frac{du}{dv}&=-\sin v=-\sin \left ( x^{2}-\pi \right ),\\ \displaystyle \frac{dv}{dx}&=2x\\ \color{black}\textrm{Seh}&\color{black}\textrm{ingga}\\ g'(x)&=\displaystyle \frac{dy}{dx}\\ &=\displaystyle \frac{dy}{du}\times \frac{du}{dv}\times \frac{dv}{dx}\\ &=\displaystyle \frac{3}{5}\cos ^{.^{-\frac{2}{5}}}\left ( x^{2}-\pi \right )\times \left ( -\sin \left ( x^{2}-\pi \right ) \right )\times (2x)\\ &=-\displaystyle \frac{6x\sin \left ( x^{2}-\pi \right )}{5\cos ^{.^{\frac{2}{5}}}\left ( x^{2}-\pi \right )}\\ &=-\displaystyle \frac{6x\sin \left ( x^{2}-\pi \right )}{5\sqrt[5]{\cos^{2} \left ( x^{2}-\pi \right )}} \end{aligned} \\\\ &\color{red}\textbf{atau kalau ingin langsungan saja}\\\\ &\color{purple}\begin{aligned}g(x)&=\sqrt[5]{\cos ^{3}\left ( x^{2}-\pi \right )}=\cos ^{.^{\frac{3}{5}}}\left ( x^{2}-\pi \right )\\ g'(x)&=\displaystyle \frac{3}{5}\cos ^{.^{-\frac{2}{5}}}\left ( x^{2}-\pi \right )\times \left ( -\sin \left ( x^{2}-\pi \right ) \right )\times (2x)\\ &=-\displaystyle \frac{6x\sin \left ( x^{2}-\pi \right )}{5\cos ^{.^{\frac{2}{5}}}\left ( x^{2}-\pi \right )}\\ &=-\displaystyle \frac{6x\sin \left ( x^{2}-\pi \right )}{5\sqrt[5]{\cos^{2} \left ( x^{2}-\pi \right )}} \end{aligned}\end{array}$

$\begin{array}{ll}\\ 3.&\textrm{Tentukan turunan pertama dari}\\ &h(x)=\cos \left ( \sin x^{2020} \right )\\\\ &\textrm{Jawab}:\\ &\color{blue}\begin{aligned}h(x)&=\cos \left ( \sin x^{2020} \right )\\ \textrm{Dim}&\textrm{isalkan}\\ y& =\cos \left ( \sin x^{2020} \right )=\cos u,\: \: \textrm{dengan}\\ u&=\sin x^{2020}=\sin v\: ,\: \textrm{serta}\: \: v=x^{2020}\\ \color{black}\textrm{mak}&\color{black}\textrm{a}\\ \displaystyle \frac{dy}{du}&=-\sin u=-\sin \left ( \sin x^{2020} \right ),\\ &\: \: \color{red}\textrm{atau}\: \: \color{black}dy=-\sin u\: \: du\\ \displaystyle \frac{du}{dv}&=\cos v\: \: \color{red}\textrm{atau}\: \: \color{black}du=\cos v\: \: dv\\ \displaystyle \frac{dv}{dx}&=2020x^{2019}\: \: \color{red}\textrm{atau}\: \: \color{black}dv=2020x^{2019}\: \: dx\\ \color{black}\textrm{Seh}&\color{black}\textrm{ingga}\\ h'(x)&=\displaystyle \frac{dy}{dx}\\ &=\displaystyle \frac{dy}{du}\times \frac{du}{dv}\times \frac{dv}{dx}\\ &=-\sin \left ( \sin x^{2020} \right )\times \cos x^{2020}\times \left ( 2020x^{2019} \right )\\ &=-2020x^{2019}\sin \left ( \sin x^{2020} \right )\cos x^{2020} \end{aligned} \\\\ &\color{black}\textbf{atau}\\ &\color{purple}\begin{aligned}dy&=-\sin u\: \: du\\ &=-\sin u\times \cos v\: \: dv\\ &=-\sin u\times \cos v\times \left ( 2020x^{2019} \right )\: \: dx\\ \displaystyle \frac{dy}{dx}&=-\sin u\times \cos v\times \left ( 2020x^{2019} \right )\\ &=.......(\color{black}\textrm{tinggal dimasukkan}) \end{aligned} \end{array}$

DAFTAR PUSTAKA

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