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微積分/微分/微分基礎/解題

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最新審查版本是已檢查在2021年6月1日。有 1個待審查更改。

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1.

f(x)=x^{2}\,

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {(x+\Delta x)^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {x^{2}+2x\Delta x+\Delta x^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x\Delta x+\Delta x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}2x+\Delta x\\&=\mathbf {2x} \end{aligned}}

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {(x+\Delta x)^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {x^{2}+2x\Delta x+\Delta x^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x\Delta x+\Delta x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}2x+\Delta x\\&=\mathbf {2x} \end{aligned}}

2.

f(x)=2x+2\,

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {[2(x+\Delta x)+2]-(2x+2)}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x+2\Delta x+2-2x-2}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2\Delta x}{\Delta x}}\\&=\mathbf {2} \end{aligned}}

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {[2(x+\Delta x)+2]-(2x+2)}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x+2\Delta x+2-2x-2}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2\Delta x}{\Delta x}}\\&=\mathbf {2} \end{aligned}}

3.

f(x)={\frac {1}{2}}x^{2}\,

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\frac {1}{2}}(x+\Delta x)^{2}-{\frac {1}{2}}x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {{\frac {1}{2}}(x^{2}+2x\Delta x+\Delta x^{2})-{\frac {1}{2}}x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {{\frac {x^{2}}{2}}+{\frac {2x\Delta x}{2}}+{\frac {\Delta x^{2}}{2}}-{\frac {x^{2}}{2}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x\Delta x+\Delta x^{2}}{2\Delta x}}\\&=\lim _{\Delta x\to 0}x+{\frac {\Delta x}{2}}\\&=\mathbf {x} \end{aligned}}

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\frac {1}{2}}(x+\Delta x)^{2}-{\frac {1}{2}}x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {{\frac {1}{2}}(x^{2}+2x\Delta x+\Delta x^{2})-{\frac {1}{2}}x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {{\frac {x^{2}}{2}}+{\frac {2x\Delta x}{2}}+{\frac {\Delta x^{2}}{2}}-{\frac {x^{2}}{2}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x\Delta x+\Delta x^{2}}{2\Delta x}}\\&=\lim _{\Delta x\to 0}x+{\frac {\Delta x}{2}}\\&=\mathbf {x} \end{aligned}}

4.

f(x)=2x^{2}+4x+4\,

5.

f(x)={\sqrt {x+2}}\,

{\displaystyle {\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\sqrt {x+\Delta x+2}}-{\sqrt {x+2}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}({\frac {{\sqrt {x+\Delta x+2}}-{\sqrt {x+2}}}{\Delta x}})({\frac {{\sqrt {x+\Delta x+2}}+{\sqrt {x+2}}}{{\sqrt {x+\Delta x+2}}+{\sqrt {x+2}}}})\\&=\lim _{\Delta x\to 0}{\frac {x+\Delta x+2-x-2}{\Delta x({\sqrt {x+\Delta x+2}}+{\sqrt {x+2}})}}\\&=\lim _{\Delta x\to 0}{\frac {\Delta x}{\Delta x({\sqrt {x+\Delta x+2}}+{\sqrt {x+2}})}}\\&=\lim _{\Delta x\to 0}{\frac {1}{{\sqrt {x+\Delta x+2}}+{\sqrt {x+2}}}}\\&

{\displaystyle {\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\sqrt {x+\Delta x+2}}-{\sqrt {x+2}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}({\frac {{\sqrt {x+\Delta x+2}}-{\sqrt {x+2}}}{\Delta x}})({\frac {{\sqrt {x+\Delta x+2}}+{\sqrt {x+2}}}{{\sqrt {x+\Delta x+2}}+{\sqrt {x+2}}}})\\&=\lim _{\Delta x\to 0}{\frac {x+\Delta x+2-x-2}{\Delta x({\sqrt {x+\Delta x+2}}+{\sqrt {x+2}})}}\\&=\lim _{\Delta x\to 0}{\frac {\Delta x}{\Delta x({\sqrt {x+\Delta x+2}}+{\sqrt {x+2}})}}\\&=\lim _{\Delta x\to 0}{\frac {1}{{\sqrt {x+\Delta x+2}}+{\sqrt {x+2}}}}\\&

6.

f(x)={\frac {1}{x}}\,

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\frac {1}{x+\Delta x}}-{\frac {1}{x}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {\frac {x-x-\Delta x}{x(x+\Delta x)}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {-\Delta x}{x\Delta x(x+\Delta x)}}\\&=\lim _{\Delta x\to 0}{\frac {-1}{x(x+\Delta x)}}\\&=\mathbf {-{\frac {1}{x^{2}}}} \end{aligned}}

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\frac {1}{x+\Delta x}}-{\frac {1}{x}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {\frac {x-x-\Delta x}{x(x+\Delta x)}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {-\Delta x}{x\Delta x(x+\Delta x)}}\\&=\lim _{\Delta x\to 0}{\frac {-1}{x(x+\Delta x)}}\\&=\mathbf {-{\frac {1}{x^{2}}}} \end{aligned}}

7.

f(x)={\frac {3}{x+1}}\,

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\frac {3}{x+\Delta x+1}}-{\frac {3}{x+1}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {\frac {3x+3-(3x+3\Delta x+3)}{(x+1)(x+\Delta x+1)}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {-3\Delta x}{\Delta x(x+1)(x+\Delta x+1)}}\\&=\lim _{\Delta x\to 0}{\frac {-3}{(x+1)(x+\Delta x+1)}}\\&=\mathbf {\frac {-3}{(x+1)^{2}}} \end{aligned}}

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\frac {3}{x+\Delta x+1}}-{\frac {3}{x+1}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {\frac {3x+3-(3x+3\Delta x+3)}{(x+1)(x+\Delta x+1)}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {-3\Delta x}{\Delta x(x+1)(x+\Delta x+1)}}\\&=\lim _{\Delta x\to 0}{\frac {-3}{(x+1)(x+\Delta x+1)}}\\&=\mathbf {\frac {-3}{(x+1)^{2}}} \end{aligned}}

8.

f(x)={\frac {1}{\sqrt {x+1}}}\,

{\begin{aligned}f'(x)&=\lim _{\Delta x\to 0}{\frac {{\frac {1}{\sqrt {x+\Delta x+1}}}-{\frac {1}{\sqrt {x+1}}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {\frac {{\sqrt {x+1}}-{\sqrt {x+\Delta x+1}}}{{\sqrt {x+\Delta x+1}}{\sqrt {x+1}}}}{\Delta x}}\\&=\lim _{\Delta x\to 0}({\frac {{\sqrt {x+1}}-{\sqrt {x+\Delta x+1}}}{\Delta x{\sqrt {x+\Delta x+1}}{\sqrt {x+1}}}})({\frac {{\sqrt {x+1}}+{\sqrt {x+\Delta x+1}}}{{\sqrt {x+1}}+{\sqrt {x+\Delta x+1}}}})\\&=\lim _{\Delta x\to 0}{\frac {x+1-(x+\Delta x+1)}{\Delta x{\sqrt {x+\Delta x+1}}{\sqrt {x+1}}({\sqrt {x+\Delta x+1}}+{\sqrt {x+1}})}}\\&=\lim _{\Delta x\to 0}{\frac {-1}{{\sqrt {x+\Delta x+1}}{\sqrt {x+1}}({\sqrt {x+\Delta x+1}}+{\sqrt {x+1}})}}\\&={\frac {-1}{(x+1)(2{\sqrt {x+1}})}}\\&=\mathbf {\frac {-1}{2(x+1)^{3/2}}} \end{aligned}}

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