加法與乘積
那些熟悉的法則不是公理——它們是定理，每一條都能從差商和極限的代數運算證出。加法法則是顯然的，因為極限可加。乘積法則需要一個巧妙的步驟：加上又減去同一項，把一個棘手的商拆成兩個好處理的部分。
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). ∎
商與鏈式
商法則遵循同樣的套路，只要你知道在 g(a) ≠ 0 處 1/g 可微（g 的連續性使得附近的 g(a + h) 不為 0）。鏈式法則才是深刻的那個。樸素的證明會乘以又除以 g(x) − g(a)——但即便 g′(a) ≠ 0，這個量也可能無窮多次為零，於是樸素的步驟並非良定義的。誠實的補救用的是一個修正函數的論證。
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). ∎