\((x^n)'=n\cdot x^{n-1}\) \((x^2)'=2x^{2-1}=2x\) \((x)'=x^{1-1}=x^0=1\) \(\left(\dfrac{1}{x}\right)'=(x^{-1})'=-1\cdot x^{-2}=-\dfrac{1}{x^2}\) \((\sqrt{x})'=(x^{\frac{1}{2}})'=\dfrac{1}{2}x^{-\frac{1}{2}}=\dfrac{1}{2}\cdot\dfrac{1}{\sqrt{x}}=\,\,\,\,\,\,=\dfrac{1}{2\sqrt{x}}\) \((\sqrt[3]{x})'=(x^{\frac{1}{3}})'=\dfrac{1}{3}x^{\frac{1}{3}-1}=\dfrac{1}{3}x^{-\frac{2}{3}}=\,\,\,\,\,\,=\dfrac{1}{3}\cdot \dfrac{1}{\sqrt[3]{x^2}}=\dfrac{1}{3\sqrt[3]{x^2}}\) \((x\cdot\sqrt{x})'= (x^1\cdot x^\frac{1}{2})^\prime=(x^{1+\frac{1}{2}})^\prime=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=(x^{\frac{3}{2}})'=\dfrac{3}{2}x^{\frac{1}{2}}=\dfrac{3}{2}\sqrt{x}\)
| \((e^x)'=e^x\) \((\ln x)'=\dfrac{1}{x}\) \((a^x)'=a^x\cdot\ln a\) \((\log_a x)'=\dfrac{1}{x\cdot\ln a}\) \((e^{ax+b})'=e^{ax+b}\cdot a\) \((\ln (ax+b))' = \dfrac{1}{ax+b}\cdot a\)
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