# Таблица производных функций

1. C′=0

2. x′=1

3. (xn)′=n·xn-1

4. ${\left( {\frac{1}{{{x^n}}}} \right)^\prime } = {\left( {{x^{ — n}}} \right)^\prime } = — n \cdot {x^{ — n — 1}} = \frac{{ — n}}{{{x^{n + 1}}}}$

5. ${\left( {\root n \of x } \right)^\prime } = {\left( {{x^{\frac{1}{n}}}} \right)^\prime } = \frac{1}{n} \cdot {x^{\frac{1}{n} — 1}} = \frac{1}{{n \cdot \root n \of {{x^{n — 1}}} }}$

6. ${\left( {\sqrt x } \right)^\prime } = \frac{1}{{2 \cdot \sqrt x }}$

7. ${\left( {\frac{1}{{\root n \of x }}} \right)^\prime } = \frac{{ — 1}}{{n \cdot \root n \of {{x^{n + 1}}} }}$

8. (ex)′= ex

9. (ax)′= ax·lna

10. (sin(x))′= cos(x)

11. (cos(x))′= –sin(x)

12. ${\left( {tg\left( x \right)} \right)^\prime } = \frac{1}{{{{\cos }^2}\left( x \right)}}$

13. ${\left( {ctg\left( x \right)} \right)^\prime } = \frac{{ — 1}}{{{{\sin }^2}\left( x \right)}}$

14. (sh(x))′= ch(x)

15. (ch(x))′= sh(x)

16. ${\left( {th\left( x \right)} \right)^\prime } = \frac{1}{{c{h^2}\left( x \right)}}$

17. ${\left( {cth\left( x \right)} \right)^\prime } = \frac{{ — 1}}{{s{h^2}\left( x \right)}}$

18. ${\left( {\ln \left( x \right)} \right)^\prime } = \frac{1}{x}$

19. ${\left( {{{\log }_a}\left( x \right)} \right)^\prime } = \frac{1}{{x\ln \left( a \right)}}$

20. ${\left( {\arcsin \left( x \right)} \right)^\prime } = \frac{1}{{\sqrt {1 — {x^2}} }}$

21. ${\left( {\arccos \left( x \right)} \right)^\prime } = \frac{{ — 1}}{{\sqrt {1 — {x^2}} }}$

22. ${\left( {arctg\left( x \right)} \right)^\prime } = \frac{1}{{1 + {x^2}}}$

23. ${\left( {arcctg\left( x \right)} \right)^\prime } = \frac{{ — 1}}{{1 + {x^2}}}$

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