Matematika Turunan dari fungsi f(x) terhadap x di titik x= ℎ didefinisikan sebagai​

valentinahesselulv – March 31, 2023

Turunan dari fungsi f(x) terhadap x di titik x= ℎ didefinisikan sebagai​

Jawaban:

1. f'(x) = 0
2. f'(x) = 6x
3. f'(x) = 6x²

Pembahasan

Turunan

Nomor 1

[tex]\large\text{$\begin{aligned}f(x)&=a\\\\f'(x)&=\lim_{h\to0}\:\frac{f(x+h)-f(x)}{h}\\&=\lim_{h\to0}\:\frac{a-a}{h}\ =\ \lim_{h\to0}\:\frac{0}{h}\\&=\lim_{h\to0}\:0\\&=\bf0\end{aligned}$}[/tex]

Penjelasan tambahan:

f(x) adalah fungsi yang bernilai konstan (hasilnya berupa sebuah konstanta, yaitu a). Berapapun nilai x, f(x) selalu menghasilkan a.

Oleh karena itu, f(x+h) juga bernilai a.

Nomor 2

[tex]\large\text{$\begin{aligned}f(x)&=3x^2\\\\f'(x)&=\lim_{h\to0}\:\frac{f(x+h)-f(x)}{h}\\&=\lim_{h\to0}\:\frac{3(x+h)^2-3x^2}{h}\\&=\lim_{h\to0}\:\frac{3(x^2+2hx+h^2)-3x^2}{h}\\&=\lim_{h\to0}\:\frac{3x^2+6hx+3h^2-3x^2}{h}\\&=\lim_{h\to0}\:\frac{6hx+3h^2}{h}\\&=\lim_{h\to0}\:\frac{\cancel{h}(6x+3h)}{\cancel{h}}\\&=\lim_{h\to0}\:6x+3h\\&=6x+0\\&=\bf6x\end{aligned}$}[/tex]

Nomor 3

[tex]\large\text{$\begin{aligned}f(x)&=2x^3\\\\f'(x)&=\lim_{h\to0}\:\frac{f(x+h)-f(x)}{h}\\&=\lim_{h\to0}\:\frac{2(x+h)^3-2x^3}{h}\\&=\lim_{h\to0}\:\frac{2(x^3+3x^2h+3xh^2+h^3)-2x^3}{h}\\&=\lim_{h\to0}\:\frac{2x^3+6x^2h+6xh^2+2h^3-2x^3}{h}\\&=\lim_{h\to0}\:\frac{6x^2h+6xh^2+2h^3}{h}\\&=\lim_{h\to0}\:\frac{\cancel{h}(6x^2+6xh+2h^2)}{\cancel{h}}\\&=\lim_{h\to0}\:6x^2+6xh+2h^2\\&=6x^2+6x(0)+2(0^2)\\&=\bf6x^2\end{aligned}$}[/tex]

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