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大家好

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這影片中我們會去看看一些常見函數

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的導數

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我們不會證明它們

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但最少我們會知道它們的導數是什麽

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好吧首先我們先看三角函數

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如果我們把sin x對x微分

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將會是cos x

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如果你看看圖像，將會更容易理解

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重申一次我們不會在這證明它們

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但有必要知道它們是什麽

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好吧sin x的導數是cos x

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那cos x呢？

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cos x對x微分會是？

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是負sin x

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sin的導數是cos

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cos的導數是 -sin

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最後tan呢？

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會是 1/ (cos^2 x)

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即 sec^2 x

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這些都很重要哦

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好吧再看看指數

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和對數

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e^x 的導數

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是微分中最酷的一個結果之一

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要知道 e 究竟有多重要

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e^x 對x微分

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我們要一點背景音樂來迎接這結果

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因為不但是微積分，也可能是數學中最酷的結果之一

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e^x的導數是e^x

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這個答案告訴我們什麽呢？

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等一下

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接下來將很刺激

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如果我們把e^x畫在圖上

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這是y軸

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x軸

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如果我們有個非常負的x值

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負得十分接近 0

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亦即e^0，即是1

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所以這裹是 1

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大約是這樣

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指數一樣

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增加得

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十分十分十分快

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這是 y= e^x的圖

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這告訴我們的是無論在哪一點

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例如這裹

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當x = 0，e^0

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即 1，在這裹的切線的斜線是？

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將會是 1

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很神奇吧

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如果 x = 1

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即 e^1，即e

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這裹的切線的斜率是？

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是e

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在不論何處

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它的切線的斜率都是函數的值

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這十分神奇

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這就是e的神奇之處

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不過這不是這影片的重點

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這影片是要告訴你常見函數的

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導數

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好吧最後了

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自然對數（Natural log）（ln）

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對x微分會是

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同樣的神奇

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這會是 1/x，或x的負一次方

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所以 ln

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可以算是填補了

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積法則

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的空隙

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即有什麽函數的導數會是

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1/x？

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冪法則告訴我們有一些函數的導數

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會是 x 的負二次方，負三次方等

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或x 的二次方，五次方等

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但留白了 x 的負一次方這個導數

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而 ln x的導數就填補了它

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現在我們沒有證明它

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我只是列表形式的列了它們出來

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在未來的影片中我們會用到它們

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並會證明它們

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（這課很輕鬆，只是把五個函數的導數背下來就可以囉！ Translated by R）