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Engineering Mathematics 微積分複習

Engineering Mathematics 微積分複習

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1<br />

證 明 (ln x)'<br />

=<br />

x<br />

令 y = ln x 等 號 兩 側 同 時 取<br />

e<br />

e<br />

d<br />

dx<br />

d<br />

dy<br />

e<br />

y<br />

y<br />

y<br />

dy<br />

dx<br />

=<br />

=<br />

e<br />

e<br />

y<br />

e<br />

y<br />

x<br />

ln x<br />

;<br />

則<br />

d<br />

= x ;<br />

dx<br />

dy d<br />

⋅ =<br />

dx dx<br />

dy<br />

⋅ = 1;<br />

dx<br />

1<br />

= ; 又<br />

y<br />

e<br />

∴ 最 後 可 得<br />

等 號 兩 側 同 時 對<br />

將<br />

y<br />

d<br />

dx<br />

依 Chain<br />

x ;<br />

e<br />

ln<br />

y<br />

微 分<br />

rule 可 寫 成<br />

此 式 簡 化 後 可 得<br />

移 至 等 號 另 一 側<br />

= ln x ; 而 e<br />

x<br />

=<br />

1<br />

x<br />

y<br />

x<br />

=<br />

x<br />

e<br />

故 得 證<br />

2005/10/7 <strong>Engineering</strong> <strong>Mathematics</strong> calculus 6

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