Question

misalkan s(x) = [tex]sin^2(x\sqrt[3]{5x-\frac{1}{x}})[/tex]. Turunan pertama dari s(x) adalah?

Answer 1

[tex]\begin{aligned}&\textsf{Turunan pertama dari }S(x)\ \sf adalah\\&S'(x)=\sin\left(2x\sqrt[3]{5x-\frac{1}{x}}\right)\left(\sqrt[3]{5x-\frac{1}{x}}+\frac{x\left(5+\dfrac{1}{x^2}\right)}{3\sqrt[3]{\left(5x-\dfrac{1}{x}\right)^2}}\right)\\\end{aligned}[/tex]
(Mirip dengan opsi D, namun berbeda. Pada opsi D, penyebut pecahannya tidak dikalikan 3.)
 

Penjelasan dengan langkah-langkah:

Turunan

Diberikan:
[tex]\begin{aligned}S(x)&=\sin^2\left(x\sqrt[3]{5x-\frac{1}{x}}\right)\\\end{aligned}[/tex]

Turunan pertama S(x):

[tex]\begin{aligned}S'(x)&=\left(\sin^2\left(x\sqrt[3]{5x-\frac{1}{x}}\right)\right)'\\&=\left(\left(\sin\left(x\sqrt[3]{5x-\frac{1}{x}}\right)\right)^2\right)'\\&\quad...\ \textsf{Aturan rantai turunan.}\\&=2\sin\left(x\sqrt[3]{5x-\frac{1}{x}}\right)\cdot\left(\sin\left(x\sqrt[3]{5x-\frac{1}{x}}\right)\right)'\end{aligned}[/tex]
[tex]\begin{aligned}S'(x)&=2\sin\left(x\sqrt[3]{5x-\frac{1}{x}}\right)\cos\left(x\sqrt[3]{5x-\frac{1}{x}}\right)\cdot\left(x\sqrt[3]{5x-\frac{1}{x}}\right)'\\&\quad...\ 2\sin\alpha\cos\alpha=\sin2\alpha\\&=\sin\left(2x\sqrt[3]{5x-\frac{1}{x}}\right)\left(x\sqrt[3]{5x-\frac{1}{x}}\right)'\\&=\sin\left(2x\sqrt[3]{5x-\frac{1}{x}}\right)\left(x\left(5x-\frac{1}{x}\right)^{1/3}\right)'\end{aligned}[/tex]
[tex]\begin{aligned}S'(x)&=\sin\left(2x\sqrt[3]{5x-\frac{1}{x}}\right)\left((x)'\left(5x-\frac{1}{x}\right)^{1/3}+\left(\left(5x-\frac{1}{x}\right)^{1/3}\right)'x\right)'\\&=\sin\left(2x\sqrt[3]{5x-\frac{1}{x}}\right)\left(\left(5x-\frac{1}{x}\right)^{1/3}+x\cdot\frac{1}{3}\left(5x-\frac{1}{x}\right)^{-2/3}\left(5x-\frac{1}{x}\right)'\right)\end{aligned}[/tex]
[tex]\begin{aligned}&=\sin\left(2x\sqrt[3]{5x-\frac{1}{x}}\right)\left(\left(5x-\frac{1}{x}\right)^{1/3}+x\cdot\frac{1}{3}\left(5x-\frac{1}{x}\right)^{-2/3}\left(5+\frac{1}{x^2}\right)\right)\\S'(x)&=\sin\left(2x\sqrt[3]{5x-\frac{1}{x}}\right)\left(\sqrt[3]{5x-\frac{1}{x}}+\frac{x\left(5+\dfrac{1}{x^2}\right)}{3\sqrt[3]{\left(5x-\dfrac{1}{x}\right)^2}}\right)\\\end{aligned}[/tex]