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Jika g(x) = a(x) × b(x) maka turunan dari g(x) adalah ..
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[tex] \tt \: a. \: g'(x) = a'(x) \: b'(x)[/tex]
[tex] \tt \: b. \: g'(x) = a'(x) \times a(x) + b'(x) \times b(x)[/tex]
[tex] \tt \: c. \: g'(x) = a'(x) \times a(x) + b'(x) \times b(x)[/tex]
[tex] \tt \: d. \: g'(x) = a'(x) \times b(x) + a(x) \times b'(x)[/tex]
[tex] \tt \: e. \: g'(x) = a'(x) \times a(c) + b'(x) \times b(c)[/tex]
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Turunan dari g(x) adalah:
[tex]\large\text{$\begin{aligned}g'(x)=\tt a'(x)\times b(x)\ +\ a(x)\times b'(x)\end{aligned}$}[/tex]
(opsi d)
Pembahasan
Turunan: Aturan Rantai Perkalian
[tex]\begin{aligned}&f(x) = a(x)\times b(x)\\\\&\textsf{Berdasarkan {de}finisi: }f'(x)\\&{=\ }\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\\\\&{=\ }\lim_{h\to0}\frac{a(x+h)\cdot b(x+h)-a(x)\cdot b(x)}{h}\end{aligned}[/tex]
[tex]\begin{aligned}&{\quad}\textsf{...tambahkan $-a(x+h)\cdot b(x)+a(x+h)\cdot b(x)$}\\&{\quad}\textsf{...karena $-a(x+h)\cdot b(x)+a(x+h)\cdot b(x)=0$}\\\\&{=\ }\lim_{h\to0}\frac{\left[\begin{array}{c}a(x+h)\cdot b(x+h)\\-\ a(x+h)\cdot b(x)+\ a(x+h)\cdot b(x)\\-\ a(x)\cdot b(x)\end{array}\right]}{h}\end{aligned}[/tex]
[tex]\begin{aligned}&{\quad}\textsf{...pisahkan}\\\\&{=\ }\lim_{h\to0}\left[\begin{array}{c}\dfrac{a(x+h)\cdot b(x+h)-a(x+h)\cdot b(x)}{h}\\\\+\ \dfrac{a(x+h)\cdot b(x)-a(x)\cdot b(x)}{h}\end{array}\right]\end{aligned}[/tex]
[tex]\begin{aligned}&{\quad}\textsf{...sifat asosiatif aljabar}\\\\&{=\ }\lim_{h\to0}\left[\begin{array}{c}a(x+h)\left(\dfrac{b(x+h)-\cdot b(x)}{h}\right)\\\\+\ \left(\dfrac{a(x+h)-a(x)}{h}\right)b(x)\end{array}\right]\end{aligned}[/tex]
[tex]\begin{aligned}&{\quad}\textsf{...sifat penjumlahan pada limit fungsi real}\\\\&{=\ }\lim_{h\to0}\left[a(x+h)\left(\frac{b(x+h)-\cdot b(x)}{h}\right)\right]\\\\&{\quad}+\lim_{h\to0}\left [\left(\frac{a(x+h)-a(x)}{h}\right)b(x)\right]\end{aligned}[/tex]
[tex]\begin{aligned}&{\quad}\textsf{...sifat perkalian pada limit fungsi real}\\&{=\ }\lim_{h\to0}a(x+h)\cdot\overbrace{\lim_{h\to0}\left(\frac{b(x+h)-\cdot b(x)}{h}\right)}^{\begin{array}{c}b'(x)\end{array}}\\\\&{\quad}+\underbrace{\lim_{h\to0}\left(\frac{a(x+h)-a(x)}{h}\right)}_{\begin{array}{c}a'(x)\end{array}}\cdot\lim_{h\to0}b(x)\\\\&{=\ }a(x+0)\cdot b'(x)\ +\ a'(x)\cdot b(x)\\\\&{=\ }\boxed{\ \tt a'(x)\times b(x)\ +\ a(x)\times b'(x)\ }\\\end{aligned}[/tex]
Penjelasan dengan langkah-langkah:
(f(x) . g(x))' = f'(x) . g(x) + f(x) . g'(x)
g(x) = a(x) . b(x)
g'(x) = a'(x) . b(x) + a(x) . b'(x)
[answer.2.content]
