Turunan Fungsi Perkalian Konstanta


Sifat 1.

Jika k suatu konstanta dan f suatu fungsi yang terdiferensial maka (kf)’ = k f'(x) yakni Dx[k f(x)] = k Dx[f(x)]

Bukti.

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

F'(x) = \lim\sb{h \to 0} \dfrac{F(x+h)-F(x)}{h}

= \lim\sb{h \to 0} \dfrac{k \cdot f(x+h)- k \cdot f(x)}{h}

= \lim\sb{h \to 0} k \cdot \dfrac{F(x+h)-F(x)}{h}

= k . \lim\sb{h \to 0} \dfrac{F(x+h)-F(x)}{h}

F'(x) = k f'(x) \blacksquare

Contoh 2.

Carilah turunan dari fungsi F(x) = 4x2

F'(x) = \lim\sb{h \to 0} \dfrac{F(x+h)-F(x)}{h}

= \lim\sb{h \to 0} \dfrac{4 \cdot (x+h)^2- k \cdot (x)^2}{h}

= \lim\sb{h \to 0} 4 \cdot \dfrac{(x+h)^2-(x)^2}{h}

= 4 . \lim\sb{h \to 0} \dfrac{(x^2+2xh+h^2)-x^2}{h}

= 4 . \lim\sb{h \to 0} \dfrac{2xh+h^2}{h}

= 4 . \lim\sb{h \to 0} (2x + h)

= 4.2x

     NOTE : f'(x) = 2x dengan f(x) = x2 (buktikan sendiri) dan k = 4

Contoh 3.

Carilah turunan dari fungsi F(x) = 2x3

F'(x) = \lim\sb{h \to 0} \dfrac{F(x+h)-F(x)}{h}

= \lim\sb{h \to 0} \dfrac{2 \cdot (x+h)^3- k \cdot (x)^3}{h}

= \lim\sb{h \to 0} 2 \cdot \dfrac{(x+h)^3-(x)^3}{h}

= 2 . \lim\sb{h \to 0} \dfrac{(x^3+3x^2h+3xh^2 +h^3)-x^3}{h}

= 2. \lim\sb{h \to 0} \dfrac{(3x^2h+3xh^2 +h^3}{h}

= 2. \lim\sb{h \to 0} (3x2 + 3xh)

= 2.3x2

NOTE : f'(x) = 3x2 dengan f(x) = x3 (buktikan sendiri) dan k = 2

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