Kalkulus/Daftar turunan

Aturan umum

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

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

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

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

$[f(g(x)){]}^{\prime }=f^{\prime }(g(x))g^{\prime }(x)$

Polinomial

• $\frac{d}{dx}(c)=0$
• $\frac{d}{dx}x=1$
• $\frac{d}{dx}{x}^n=n{x}^{n-1}$
• $\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}$
• $\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$
• $\frac{d}{dx}(c_n{x}^n+c_{n-1}{x}^{n-1}+c_{n-2}{x}^{n-2}+\cdots +c_2{x}^2+c_{1}x+c_0)=n{c}_n{x}^{n-1}+(n-1)c_{n-1}{x}^{n-2}+(n-2)c_{n-2}{x}^{n-3}+\cdots +2c_{2}x+c_1$

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

• $\frac{d}{dx}{e}^x=e^x$
• $\frac{d}{dx}{a}^x=a^x\ln (a)\text{jika }a>0$
• $\frac{d}{dx}\ln (x)=\frac{1}{x}$
• $\frac{d}{dx}{\log }_a(x)=\frac{1}{x\ln (a)}\text{jika }a>0,a\ne 1$
• $(f^g{)}^{\prime }=\left(e^{g\ln f}\right)=f^g(f^{\prime }\frac{g}{f}+g^{\prime }\ln f),f>0$
• $(c^f{)}^{\prime }=\left(e^{f\ln c}\right)=f^{\prime }{c}^f\ln c$

Fungsi invers trigonometri

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

Fungsi hiperbola dan invers hiperbola

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

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

$\frac{d}{dx}\tanh x={\text{sech}}^{2}x$

$\frac{d}{dx}\text{sech}x=-\tanh x\text{sech}x$

$\frac{d}{dx}\text{coth}x=-{\text{csch}}^{2}x$

$\frac{d}{dx}\text{csch}x=-\text{coth}x\text{csch}x$

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

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

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

$\frac{d}{dx}{\text{sech}}^{-1}x=\frac{1}{x\sqrt{1-x^2}}$

$\frac{d}{dx}{\text{coth}}^{-1}x=\frac{-1}{1-x^2}$

$\frac{d}{dx}{\text{csch}}^{-1}x=\frac{-1}{|x|\sqrt{1+x^2}}$