اشتقاق (أمثلة)

الصفحة الرئيسية: إشتقاق

 مثال 1

لنعتبر f(x) = 5:

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

 مثال 2

${\displaystyle{\displaystyle f^{\prime}(4)\,}}$	${\displaystyle{\displaystyle=\lim_{h\rightarrow 0}{\frac{f(4+h)-f(4)}{h}}}}$
	${\displaystyle{\displaystyle=\lim_{h\rightarrow 0}{\frac{2(4+h)-3-(2\cdot 4-3)}{h}}}}$
	${\displaystyle{\displaystyle=\lim_{h\rightarrow 0}{\frac{8+2h-3-8+3}{h}}}}$
	${\displaystyle{\displaystyle=\lim_{h\rightarrow 0}{\frac{2h}{h}}=2}}$

 مثال 3

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

 مثال 4

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

 مثال 5

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