الرئيسيةبحث

اشتقاق (أمثلة)

الصفحة الرئيسية: إشتقاق

فهرس

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

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

f'(x)\, = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
= \lim_{h\rightarrow 0}\frac{(x+h)^2 - x^2}{h}
= \lim_{h\rightarrow 0}\frac{x^2 + 2xh + h^2 - x^2}{h}
= \lim_{h\rightarrow 0}\frac{2xh + h^2}{h}
= \lim_{h\rightarrow 0}(2x + h) = 2x


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}}