加法与乘积
那些熟悉的法则不是公理——它们是定理，每一条都能从差商和极限的代数运算证出。加法法则是显然的，因为极限可加。乘积法则需要一个巧妙的步骤：加上又减去同一项，把一个棘手的商拆成两个好处理的部分。
Product rule: if f, g are differentiable at a, then (fg)'(a) = f'(a)g(a) + f(a)g'(a). Difference quotient of fg at a (h ≠ 0): ( f(a+h)g(a+h) - f(a)g(a) ) / h Add and subtract f(a+h)g(a): = ( f(a+h)g(a+h) - f(a+h)g(a) + f(a+h)g(a) - f(a)g(a) ) / h = f(a+h) · ( g(a+h) - g(a) ) / h + ( f(a+h) - f(a) ) / h · g(a) Let h -> 0: f(a+h) -> f(a) (f is continuous at a, since differentiable) (g(a+h)-g(a))/h -> g'(a) (f(a+h)-f(a))/h -> f'(a) By algebra of limits: (fg)'(a) = f(a)·g'(a) + f'(a)·g(a). ∎
商与链式
商法则遵循同样的套路，只要你知道在 g(a) ≠ 0 处 1/g 可微（g 的连续性使得附近的 g(a + h) 不为 0）。链式法则才是深刻的那个。朴素的证明会乘以又除以 g(x) − g(a)——但即便 g′(a) ≠ 0，这个量也可能无穷多次为零，于是朴素的步骤并非良定义的。诚实的补救用的是一个修正函数的论证。
Chain rule: with g differentiable at a and f differentiable at b = g(a),
(f ∘ g)'(a) = f'(b) · g'(a).
Key device: define a function E (the 'error') near b by
E(y) = ( f(y) - f(b) ) / ( y - b ) - f'(b) for y ≠ b,
E(b) = 0.
Then f(y) - f(b) = ( f'(b) + E(y) ) · ( y - b ) -- and this holds even at y = b
(both sides are 0). Because f is differentiable at b, E(y) -> 0 as y -> b, so E is
continuous at b.
Now put y = g(x), b = g(a):
f(g(x)) - f(g(a)) = ( f'(b) + E(g(x)) ) · ( g(x) - g(a) ).
Divide by x - a (x ≠ a):
( f(g(x)) - f(g(a)) ) / (x - a)
= ( f'(b) + E(g(x)) ) · ( g(x) - g(a) ) / (x - a).
Let x -> a: g continuous so g(x) -> b, hence E(g(x)) -> E(b) = 0;
(g(x)-g(a))/(x-a) -> g'(a).
Therefore the limit is ( f'(b) + 0 ) · g'(a) = f'(g(a)) · g'(a). ∎