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d d x ( f + g ) = d f d x + d g d x {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(f+g)={\frac {\mathrm {d} f}{\mathrm {d} x}}+{\frac {\mathrm {d} g}{\mathrm {d} x}}}
d d x ( c ⋅ f ) = c ⋅ d f d x {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(c\cdot f)=c\cdot {\frac {\mathrm {d} f}{\mathrm {d} x}}}
d d x ( f ⋅ g ) = f ⋅ d g d x + g ⋅ d f d x {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(f\cdot g)=f\cdot {\frac {\mathrm {d} g}{\mathrm {d} x}}+g\cdot {\frac {\mathrm {d} f}{\mathrm {d} x}}}
d d x ( f g ) = − f ⋅ d g d x + g ⋅ d f d x g 2 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {f}{g}}\right)={\dfrac {-f\cdot {\dfrac {\mathrm {d} g}{dx}}+g\cdot {\dfrac {\mathrm {d} f}{\mathrm {d} x}}}{g^{2}}}}
d d x [ f ( g ( x ) ) ] = d f d g ⋅ d g d x = f ′ ( g ( x ) ) ⋅ g ′ ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}[f(g(x))]={\frac {\mathrm {d} f}{\mathrm {d} g}}\cdot {\frac {\mathrm {d} g}{\mathrm {d} x}}=f'(g(x))\cdot g'(x)}
d n d x n f ( x ) g ( x ) = ∑ i = 0 n ( n i ) f ( n − i ) ( x ) g ( i ) ( x ) {\displaystyle {\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}f(x)g(x)=\sum _{i=0}^{n}\left({\begin{matrix}n\\i\end{matrix}}\right)f^{(n-i)}(x)g^{(i)}(x)}
d d x ( 1 f ) = − f ′ f 2 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {1}{f}}\right)=-{\frac {f'}{f^{2}}}}
d d x sin ( x ) = cos ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sin(x)=\cos(x)}
d d x cos ( x ) = − sin ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cos(x)=-\sin(x)}
d d x tan ( x ) = sec 2 ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\tan(x)=\sec ^{2}(x)}
d d x cot ( x ) = − csc 2 ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cot(x)=-\csc ^{2}(x)}
d d x sec ( x ) = sec ( x ) tan ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sec(x)=\sec(x)\tan(x)}
d d x csc ( x ) = − csc ( x ) cot ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\csc(x)=-\csc(x)\cot(x)}
d d x arcsin ( x ) = 1 1 − x 2 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}}
d d x arccos ( x ) = − 1 1 − x 2 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}}
d d x arctan ( x ) = 1 x 2 + 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arctan(x)={\frac {1}{x^{2}+1}}}
d d x arccot ( x ) = − 1 x 2 + 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccot}(x)=-{\frac {1}{x^{2}+1}}}
d d x arcsec ( x ) = 1 | x | x 2 − 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arcsec}(x)={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d d x arccsc ( x ) = − 1 | x | x 2 − 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccsc}(x)=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d d x sinh ( x ) = cosh ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sinh(x)=\cosh(x)}
d d x cosh ( x ) = sinh ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cosh(x)=\sinh(x)}
d d x tanh ( x ) = s e c h 2 ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\tanh(x)={\rm {sech}}^{2}(x)}
d d x s e c h ( x ) = − tanh ( x ) s e c h ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {sech}}(x)=-\tanh(x){\rm {sech}}(x)}
d d x coth ( x ) = − c s c h 2 ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\coth(x)=-{\rm {csch}}^{2}(x)}
d d x c s c h ( x ) = − coth ( x ) c s c h ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {csch}}(x)=-\coth(x){\rm {csch}}(x)}
d d x a r s i n h ( x ) = 1 x 2 + 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arsinh}}(x)={\frac {1}{\sqrt {x^{2}+1}}}}
d d x a r c o s h ( x ) = 1 x 2 − 1 , x > 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcosh}}(x)={\frac {1}{\sqrt {x^{2}-1}}}\ ,\ x>1}
d d x a r t a n h ( x ) = 1 1 − x 2 , | x | < 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {artanh}}(x)={\frac {1}{1-x^{2}}}\ ,\ |x|<1}
d d x a r s e c h ( x ) = − 1 x 1 − x 2 , 0 < x < 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arsech}}(x)=-{\frac {1}{x{\sqrt {1-x^{2}}}}}\ ,\ 0<x<1}
d d x a r c o t h ( x ) = 1 1 − x 2 , | x | > 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcoth}}(x)={\frac {1}{1-x^{2}}}\ ,\ |x|>1}
d d x a r c s c h ( x ) = − 1 | x | 1 + x 2 , x ≠ 0 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcsch}}(x)=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}\ ,\ x\neq 0}
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