Turunan Fungsi Aturan Perkalian


Sifat :

Jika f dan g fungsi – fungsi yang terdiferensial maka (f . g)'(x) = f'(x)g(x) + f(x)g'(x) yakni Dx[f(x)g(x)] = Dx[f(x)]g(x) + f(x)Dx[g(x)]

Bukti :

Andaikan : F(x) = f(x).g(x)

F'(x) = lim_{h \to 0} \quad \dfrac{F(x+h)-F(x)}{h}

= lim_{h \to 0} \quad \dfrac{f(x+h).g(x+h)-f(x).g(x)}{h}

= lim_{h \to 0} \quad \dfrac{f(x+h).g(x+h)-g(x+h).f(x)+g(x+h).f(x)-f(x).g(x)}{h}

= lim_{h \to 0} \quad \dfrac{g(x+h)(f(x+h)-f(x))}{h} + \dfrac{f(x)(g(x+h)-g(x))}{h}

= lim_{h \to 0} \quad g(x+h) \dfrac{f(x+h)-f(x)}{h}+f(x+h)\dfrac{g(x+h)-g(x)}{h}

= lim_{h \to 0} \quad g(x+h) \dfrac{f(x+h)-f(x)}{h} + lim_{h \to 0} \quad f(x+h)\dfrac{g(x+h)-g(x)}{h}

= lim_{h \to 0} \quad g(x+h). lim_{h \to 0} \quad \dfrac{f(x+h)-f(x)}{h} + lim_{h \to 0} \quad f(x+h). lim_{h \to 0} \quad \dfrac{g(x+h)-g(x)}{h}

= f'(x)g(x) + g'(x)f(x) \blacksquare

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