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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