連鎖律
微分の定義より
( f ∘ g ) ′ ( a ) = lim x → a ( f ∘ g ) ( x ) − ( f ∘ g ) ( a ) x − a = lim x → a f ( g ( x ) ) − f ( g ( a ) ) x − a = lim x → a [ f ( g ( x ) ) − f ( g ( a ) ) g ( x ) − g ( a ) ⋅ g ( x ) − g ( a ) x − a ] = lim x → a f ( g ( x ) ) − f ( g ( a ) ) g ( x ) − g ( a ) ⋅ lim x → a g ( x ) − g ( a ) x − a = f ′ ( g ( a ) ) ⋅ g ′ ( a ) {\displaystyle {\begin{aligned}(f\circ g)'(a)~&=\lim _{x\rightarrow a}{(f\circ g)(x)-(f\circ g)(a) \over x-a}\\&=\lim _{x\rightarrow a}{f(g(x))-f(g(a)) \over x-a}\\&=\lim _{x\rightarrow a}\left[{f(g(x))-f(g(a)) \over g(x)-g(a)}\cdot {g(x)-g(a) \over x-a}\right]\\&=\lim _{x\rightarrow a}{f(g(x))-f(g(a)) \over g(x)-g(a)}\cdot \lim _{x\rightarrow a}{g(x)-g(a) \over x-a}\\&=f'(g(a))\cdot g'(a)\end{aligned}}}