Derivatele funcţiilor elementare

Funcţia Derivata funcţiei Domeniul derivatei
\(f(x)=c,\:c\in\mathbf{R}\) \(f'(x)=0\) \(\mathbf{R}\)
\(f(x)=x^{n},\:n\in\mathbf{N}^{*}\) \(f'(x)=nx^{n-1}\) \(\mathbf{R}\)
\(f(x)=x^{a},\:a\in\mathbf{R}\) \(f'(x)=ax^{a-1}\) \(\mathbf{R}\)
\(f(x)=\sqrt{x}\) \(f'(x)=\frac{1}{2\sqrt{x}}\) \((0,+\infty)\)
\(f(x)=\log_{a}x\) \(f'(x)=\frac{1}{x\ln a}\) \((0,+\infty)\)
\(f(x)=\ln{x}\) \(f'(x)=\frac{1}{x}\) \((0,+\infty)\)
\(f(x)=a^{x},\:a>0,\: a\neq 0\) \(f'(x)=a^{x}\ln{a}\) \(\mathbf{R}\)
\(f(x)=\sin{x}\) \(f'(x)=\cos{x}\) \(\mathbf{R}\)
\(f(x)=\cos{x}\) \(f'(x)=-\sin{x}\) \(\mathbf{R}\)
\(f(x)=tgx\) \(f'(x)=\frac{1}{\cos^{2}{x}}\) \(\mathbf{R}-\left\{ \frac{\pi}{2}+k\pi|k\in\mathbf{Z}\right\}\)
\(f(x)=\arcsin x\) \(f'(x)=\frac{1}{\sqrt[]{1-x^{2}}}\) \((-1,1)\)
\(f(x)=\arccos x\) \(f'(x)=-\frac{1}{\sqrt[]{1-x^{2}}}\) \((-1,1)\)
\(f(x)=arctg\;x\) \(f'(x)=\frac{1}{1+x^{2}}\) \(\mathbf{R}\)
\(f(x)=arcctg\;x\) \(f'(x)=-\frac{1}{1+x^{2}} \) \(\mathbf{R}\)
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