اشتقاق (أمثلة)

الصفحة الرئيسية: اشتقاق

مثال 1

لنعتبر $f(x)=5$:

$f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow 0} \frac{5-5}{h} = 0$

مثال 2

لنعتبر $f(x)=2x-3$:

$f'(4)\,$

$= \lim_{h\rightarrow 0}\frac{f(4+h)-f(4)}{h}$

$= \lim_{h\rightarrow 0}\frac{2(4+h)-3-(2\cdot 4-3)}{h}$

$= \lim_{h\rightarrow 0}\frac{8+2h-3-8+3}{h}$

$= \lim_{h\rightarrow 0}\frac{2h}{h} = 2$

مثال 3

لنعتبر $f(x) = x^2\,$ :

$f'(x)\,$

مثال 4

$f'(x)\,$	$= \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$
	$= \lim_{h\rightarrow 0}\frac{\sqrt{x+h} - \sqrt{x}}{h}$
	$= \lim_{h\rightarrow 0}\frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})}$
	$= \lim_{h\rightarrow 0}\frac{x+h - x}{h(\sqrt{x+h} + \sqrt{x})}$
	$= \lim_{h\rightarrow 0}\frac{1}{\sqrt{x+h} + \sqrt{x}}$
	$= \frac{1}{2 \sqrt{x}}$

مثال 5

$f''(x)\,$	$= \lim_{h\rightarrow 0}\frac{f'(x+h)-f'(x)}{h}$
	$= \lim_{h\rightarrow 0} \frac{\frac{1}{2 \sqrt{x+h}}-\frac{1}{2 \sqrt{x}}}{h}$
	$= \lim_{h\rightarrow 0} \frac{\left(\frac{1}{2 \sqrt{x+h}}-\frac{1}{2 \sqrt{x}}\right)(2 \sqrt{x+h}+2 \sqrt{x})}{h(2 \sqrt{x+h}+2 \sqrt{x})}$
	$= \lim_{h\rightarrow 0} \frac{\frac{2 \sqrt{x}}{2 \sqrt{x+h}}-\frac{2 \sqrt{x+h}}{2 \sqrt{x}}}{h(2 \sqrt{x+h}+2 \sqrt{x})}$
	$= \lim_{h\rightarrow 0} \frac{\frac{x}{\sqrt{x} \sqrt{x+h}}-\frac{x+h}{\sqrt{x} \sqrt{x+h}}}{h(2 \sqrt{x+h}+2 \sqrt{x})}$
	$= \lim_{h\rightarrow 0} \frac{\frac{-h}{\sqrt{x} \sqrt{x+h}}}{h(2 \sqrt{x+h}+2 \sqrt{x})}$
	$= \lim_{h\rightarrow 0} \frac{-1}{\sqrt{x} \sqrt{x+h} (2 \sqrt{x+h}+2 \sqrt{x})}$
	$= \lim_{h\rightarrow 0} \frac{-1}{2 \sqrt{x} (x+h) + 2 x \sqrt{x+h}}$
	$= \frac{-1}{4 x \sqrt{x}}$
	$= \frac{1}{4 x \sqrt{x}}$