اشتقاق (أمثلة)

الصفحة الرئيسية: إشتقاق

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 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

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