Kalkulus/Daftar turunan

Aturan umum

${\frac {d}{dx}}(f+g)={\frac {df}{dx}}+{\frac {dg}{dx}}$

${\frac {d}{dx}}(cf)=c{\frac {df}{dx}}$

${\frac {d}{dx}}(fg)=f{\frac {dg}{dx}}+g{\frac {df}{dx}}$

${\frac {d}{dx}}\left({\frac {f}{g}}\right)={\frac {g{\frac {df}{dx}}-f{\frac {dg}{dx}}}{g^{2}}}$

$[f(g(x))]'=f'(g(x))g'(x)$

Polinomial

• ${\frac {d}{dx}}(c)=0$
• ${\frac {d}{dx}}x=1$
• ${\frac {d}{dx}}x^{n}=nx^{n-1}$
• ${\frac {d}{dx}}{\sqrt {x}}={\frac {1}{2{\sqrt {x}}}}$
• ${\frac {d}{dx}}{\frac {1}{x}}=-{\frac {1}{x^{2}}}$
• ${\frac {d}{dx}}(c_{n}x^{n}+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots +c_{2}x^{2}+c_{1}x+c_{0})=nc_{n}x^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots +2c_{2}x+c_{1}$

Fungsi trigonometri

${\frac {d}{dx}}\sin(x)=\cos(x)$

${\frac {d}{dx}}\cos(x)=-\sin(x)$

${\frac {d}{dx}}\tan(x)=\sec ^{2}(x)$

${\frac {d}{dx}}\cot(x)=-\csc ^{2}(x)$

${\frac {d}{dx}}\sec(x)=\sec(x)\tan(x)$

${\frac {d}{dx}}\csc(x)=-\csc(x)\cot(x)$

Fungsi eksponensial dan logaritma

• ${\frac {d}{dx}}e^{x}=e^{x}$
• ${\frac {d}{dx}}a^{x}=a^{x}\ln(a)\qquad {\mbox{jika }}a>0$
• ${\frac {d}{dx}}\ln(x)={\frac {1}{x}}$
• ${\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}\qquad {\mbox{jika }}a>0,a\neq 1$
• $(f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\qquad f>0$
• $(c^{f})'=\left(e^{f\ln c}\right)'=f'c^{f}\ln c$

Fungsi invers trigonometri

• ${\frac {d}{dx}}{\mbox{arcsin x}}={\frac {1}{\sqrt {1-x^{2}}}}$
• ${\frac {d}{dx}}{\mbox{arccos x}}=-{\frac {1}{\sqrt {1-x^{2}}}}$
• ${\frac {d}{dx}}{\mbox{arctan x}}={\frac {1}{1+x^{2}}}$
• ${d \over dx}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}}}$
• ${d \over dx}\operatorname {arccot} x={-1 \over 1+x^{2}}$
• ${d \over dx}\operatorname {arccsc} x={-1 \over |x|{\sqrt {x^{2}-1}}}$

Fungsi hiperbola dan invers hiperbola

${d \over dx}\sinh x=\cosh x$

${d \over dx}\cosh x=\sinh x$

${d \over dx}\tanh x={\mbox{sech}}^{2}\,x$

${d \over dx}\,{\mbox{sech}}\,x=-\tanh x\,{\mbox{sech}}\,x$

${d \over dx}\,{\mbox{coth}}\,x=-\,{\mbox{csch}}^{2}\,x$

${d \over dx}\,{\mbox{csch}}\,x=-\,{\mbox{coth}}\,x\,{\mbox{csch}}\,x$

${d \over dx}\sinh ^{-1}x={1 \over {\sqrt {x^{2}+1}}}$

${d \over dx}\cosh ^{-1}x={-1 \over {\sqrt {x^{2}-1}}}$

${d \over dx}\tanh ^{-1}x={1 \over 1-x^{2}}$

${d \over dx}{\mbox{sech}}^{-1}\,x={1 \over x{\sqrt {1-x^{2}}}}$

${d \over dx}{\mbox{coth}}^{-1}\,x={-1 \over 1-x^{2}}$

${d \over dx}{\mbox{csch}}^{-1}\,x={-1 \over |x|{\sqrt {1+x^{2}}}}$