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\begin{eqnarray} (x^n)' & = & n\cdot x^{n-1},\qquad n\in\mathbb{R} \\ (\ln x)' & = & \frac{1}{x} \\ (\log_ax)' & = & \frac{1}{x\cdot\ln a}, \qquad a>0,\ a\neq1 \\ (\ee^x)' & = & \ee^x \\ (a^x)' & = & a^x\cdot \ln a,\qquad a>0 \\ (\sin x)' & = & \cos x \\ (\cos x)' & = & -\sin x \\ (\tg x)' & = & \frac{1}{\cos^2x} \\ (\cotg x)' & = & -\frac{1}{\sin^2x} \\ (\arcsin x)' & = & \frac{1}{\sqrt{1-x^2}} \\ (\arccos x)' & = & -\frac{1}{\sqrt{1-x^2}} \\ (\arctg x)' & = & \frac{1}{1+x^2} \\ (\arccotg x)' & = & -\frac{1}{1+x^2} \end{eqnarray}
\begin{eqnarray} (f\pm g)' & = & f' \pm g'\qquad \mbox{(derivace součtu a rozdílu funkcí)}\\ (f\cdot g)' & = & f'g + fg' \qquad \mbox{(derivace součinu funkcí)}\\ \left(\frac{f}{g}\right) & = & \frac{f'g-g'f}{g^2} \qquad \mbox{(derivace podílu funkcí)} \\ (f(g(x))' & = & f'(g(x)) \cdot g'(x) \qquad \mbox{(derivace složené funkce)} \\ \left(f^{-1}(x)\right)' & = & \frac{1}{f'(f^{-1}(x))} \qquad \mbox{(derivace inverzní funkce)} \end{eqnarray}
\begin{eqnarray} \left(x^7\right)' & = & \left(x^3\cdot x^4\right)' \;=\; \left(x^3\right)'\cdot x^4 + x^3\cdot \left(x^4\right)' \\ \; & = & \; 3x^2\cdot x^4 + x^3\cdot 4x^3 \; = \; 3x^6+4x^6 \; =\; 7x^6 \nonumber \\ \end{eqnarray} \begin{eqnarray} \left(\frac{x^7}{x^4}\right)' & = & \frac{\left(x^7\right)'\cdot x^4 -x^7\cdot\left(x^4\right)'}{(x^4)^2} \; = \; \frac{7x^6\cdot x^4-x^7\cdot 4x^3}{x^8} \\ \; & = & \; \frac{7x^{10}-4x^{10}}{x^8} \; = \; \frac{3x^{10}}{x^8} \; = \; 3x^2 \nonumber \\ \end{eqnarray} \begin{eqnarray} \left(x^{15}\right)' & = & \left(x^3\right)^5 \; = \; 5\left(x^3\right)^4 \cdot \left(x^3\right)' \; = \; 5x^{12}\cdot 3x^2 \; = \; 15x^{14} \\ \left(\sqrt[k]{x}\right)' & = & \frac{1}{k\cdot \left(\sqrt[k]{x}\right)^{k-1}} \; = \; \frac{1}{k\cdot x^{\frac{k-1}{k}}} \; = \; \frac{\sqrt[k]{x}}{kx} \\ \end{eqnarray} \begin{eqnarray} (\sin^2x)' & = & (\sin x\cdot \sin x)' \; = \; (\sin x)'\cdot\sin x + \sin x\cdot (\sin x)' \\ \; & = & \; \cos x\cdot\sin x+\sin x\cdot\cos x \; = \; 2\sin x\cos x \nonumber \\ \end{eqnarray} \begin{eqnarray} \left(\frac{\sin x}{\cos x}\right)' & = & \frac{(\sin x)'\cdot\cos x - \sin x\cdot (\cos x)'}{\cos^2x} \\ \; & = & \; \frac{\cos x\cdot\cos x-\sin x\cdot(-\sin x) }{\cos^2x} \nonumber \\ \; & = & \; \frac{\cos^2x+\sin^2x}{\cos^2x} \; = \; \frac{1}{\cos^2x} \nonumber \\ \end{eqnarray} \begin{eqnarray} \left(\sin x^2\right)' & = & \left(\cos x^2\right) \cdot \left(x^2\right)' \; = \; 2x\cdot\cos x^2 \\ (\arcsin x)' & = & \frac{1}{\cos(\arcsin x)} \; = \; \frac{1}{\sqrt{1-x^2}} \\ (\ln x)' & = & \frac{1}{\ee^{\ln x}} \; = \; \frac{1}{x} \end{eqnarray}
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