כלל השרשרת :
( f ( g ( x ) ) ′ = f ′ ( g ( x ) ) ⋅ g ( x ) ′ {\displaystyle (f(g(x))'=f'(g(x))\cdot g(x)'}
1 x = x − 1 f n ( x ) = n ⋅ f n − 1 ( x ) ⋅ f ′ ( x ) c f ( x ) ′ = − c ∗ f ′ ( x ) f ( x ) 2 ( f ( x ) ) ′ = f ( x ) ′ − 1 2 = − 1 2 f − 1 2 − 1 ( x ) ⋅ f ′ ( x ) = 1 2 f ( x ) ⋅ f ′ ( x ) {\displaystyle {\begin{aligned}{\frac {1}{x}}&=x^{-1}\\f^{n}(x)&=n\cdot f^{n-1}(x)\cdot f'(x)\\{\frac {c}{f(x)}}'=-{\frac {c*f'(x)}{f(x)^{2}}}\\({\sqrt {f(x)}})'=f(x)'^{-{\frac {1}{2}}}&=-{\frac {1}{2}}f^{-{\frac {1}{2}}-1}(x)\cdot f'(x)={\frac {1}{2{\sqrt {f(x)}}}}\cdot f'(x)\end{aligned}}}
sin ( f ( x ) ) ′ = cos f ( x ) ⋅ f ′ ( x ) cos ( f ( x ) ) ′ = − sin f ( x ) ⋅ f ′ ( x ) tan ( f ( x ) ) ′ = sin ( f ( x ) ) cos ( f ( x ) ) ′ = f ′ ( x ) cos 2 f ( x ) {\displaystyle {\begin{aligned}\sin(f(x))'&=\cos f(x)\cdot f'(x)\\\cos(f(x))'&=-\sin f(x)\cdot f'(x)\\\tan(f(x))'&={\frac {\sin(f(x))}{\cos(f(x))}}'={\frac {f'(x)}{\cos ^{2}f(x)}}\end{aligned}}}
arcsin f ( x ) ′ = f ( x ) ′ 1 − f ( x ) 2 arccos f ( x ) ′ = − f ( x ) ′ ) 1 − f ( x ) 2 arctan f ( x ) ′ = f ′ ( x ) 1 + f 2 ( x ) {\displaystyle {\begin{aligned}\arcsin f(x)'&={\frac {f(x)'}{\sqrt {1-f(x)^{2}}}}\\\arccos f(x)'&=-{\frac {f(x)')}{\sqrt {1-f(x)^{2}}}}\\\arctan f(x)'&={\frac {f'(x)}{1+f^{2}(x)}}\end{aligned}}}
( e f ( x ) ) ′ = e f ( x ) ⋅ f ′ ( x ) ( a f ( x ) ) ′ = a f ( x ) ⋅ ln a ⋅ f ′ ( x ) ln ( f ( x ) ) ′ = f ′ ( x ) f ( x ) log a f ( x ) ′ = f ′ ( x ) f ( x ) ⋅ ln a {\displaystyle {\begin{aligned}(e^{f(x)})'&=e^{f(x)}\cdot f'(x)\\(a^{f(x)})'&=a^{f(x)}\cdot \ln a\cdot f'(x)\\\ln(f(x))'&={\frac {f'(x)}{f(x)}}\\\log _{a}f(x)'&={\frac {f'(x)}{f(x)\cdot \ln a}}\end{aligned}}}
