导数
对于函数 (fleft(x ight))，定义其导数为 (f'left(x ight))。
(f'left(x ight)) 可以看作 (fleft(x ight)) 在相应位置的斜率。
多项式函数的求导
对于一个多项式函数 (fleft(x ight) = sum_{i=0} a_i x^i)，这个函数的导数为 (f'left(x ight) = sum_{i = 0} i cdot a_i x ^ {i - 1})（系数乘上指数，指数上减一）。
特殊函数导数
[left(sinleft(x
ight)
ight)' = cosleft(x
ight)
]
[left(cosleft(x
ight)
ight)' = -sinleft(x
ight)
]
[left(lnleft(x
ight)
ight)' = dfrac{1}{x}
]
[ ext{指数函数：}left(a^x
ight)' = a^xlnleft(a
ight) ext{（将会在后文给出证明）}
]
[ ext{特别的：}left(e^x
ight)' = e^xlnleft(e
ight) = e^x cdot 1 = e^x
]
一般函数的求导
对于一个一般函数 (fleft(x ight))，其在 (x) 处的导数为
[lim_{Delta x o 0} dfrac{fleft(x + Delta x
ight) - fleft(x
ight)}{Delta x}
]
和积商求导法则
和差
[left(fleft(x
ight) pm gleft(x
ight)
ight)' = f'left(x
ight) pm g'left(x
ight)
]
证明略
积
[left(fleft(x
ight)gleft(x
ight)
ight)' = f'left(x
ight)gleft(x
ight) + fleft(x
ight)g'left(x
ight)
]
证明：
[left(fleft(x
ight)gleft(x
ight)
ight)' = lim_{Delta x o 0} dfrac{fleft(x + Delta x
ight)gleft(x + Delta x
ight) - fleft(x
ight)gleft(x
ight)}{Delta x}
]
[= lim_{Delta x o 0} dfrac{left(fleft(x + Delta x
ight)gleft(x + Delta x
ight) - fleft(x
ight)gleft(x + Delta x
ight)
ight) + left(fleft(x
ight)gleft(x + Delta x
ight) - fleft(x
ight)gleft(x
ight)
ight)}{Delta x}
]
[= lim_{Delta x o 0} dfrac{fleft(x + Delta x
ight) - fleft(x
ight)}{Delta x}gleft(x + Delta x
ight) + fleft(x
ight)lim_{Delta x o 0}dfrac{gleft(x + Delta x
ight) - gleft(x
ight)}{Delta x}
]
[= f'left(x
ight)gleft(x
ight) + fleft(x
ight)g'left(x
ight)
]
商
[left(dfrac{fleft(x
ight)}{gleft(x
ight)}
ight)' = dfrac{f'left(x
ight)gleft(x
ight) - fleft(x
ight)g'left(x
ight)}{g^2left(x
ight)}
]
证明：
[left(dfrac{fleft(x
ight)}{gleft(x
ight)}
ight)' = lim_{Delta x o 0} dfrac{dfrac{fleft(x + Delta x
ight)}{gleft(x + Delta x
ight)} - dfrac{fleft(x
ight)}{gleft(x
ight)}}{Delta x}
]
[= lim_{Delta x o 0} dfrac{fleft(x + Delta x
ight)gleft(x
ight) - gleft(x + Delta x
ight)fleft(x
ight)}{gleft(x +Delta x
ight)gleft(x
ight)Delta x}
]
[= lim_{Delta x o 0} dfrac{left(fleft(x + Delta x
ight)gleft(x
ight) - fleft(x
ight)gleft(x
ight)
ight) - left(gleft(x + Delta x
ight)fleft(x
ight) - fleft(x
ight)gleft(x
ight)
ight)}{gleft(x +Delta x
ight)gleft(x
ight)Delta x}
]
[= lim_{Delta x o 0} dfrac{dfrac{fleft(x + Delta x
ight) - fleft(x
ight)}{Delta x}gleft(x
ight) - dfrac{gleft(x + Delta x
ight) - gleft(x
ight)}{Delta x}fleft(x
ight)}{gleft(x +Delta x
ight)gleft(x
ight)}
]
[= dfrac{f'left(x
ight)gleft(x
ight) - fleft(x
ight)g'left(x
ight)}{g^2left(x
ight)}
]
复合函数
[left(fleft(gleft(x
ight)
ight)
ight)' = f'left(gleft(x
ight)
ight)g'left(x
ight)
]
证明：
咕
其他
那么，关于前文“特殊函数求导”的指数函数求导，可以简单给出证明了。
[ ext{设} y = a^x
]
[ ext{则有} lnleft(y
ight) = xlnleft(a
ight)
]
[left(lnleft(y
ight)
ight)' = left(xlnleft(a
ight)
ight)'
]
[dfrac{y'}{y} = 1 cdot lnleft(a
ight) + 0
]
（这一步左边使用了复合函数求导法则，右边使用了积的求导法则。）
再将右边的分母上的 (y) 乘到右边，并替换回 (a^x)，就得到了：
[y' = a^xlnleft(a
ight)
]