# Derivative formulas

**Calculation of the derivative**— the most important operation in differential calculus.

## Functions differentiation formula

In the table below u and v — are functions of the variable x, and c — is constant. It is enough these formulas to differentiate any elementary function.

(c · u)′ = c · u ′

(u + v)′ = u ′ + v ′

(u · v)′ = u ′ · v + u · v ′

( | u | ) | ′ | = | u ′ · v - u · v ′ |

v | v^{2} |

## Basic formulas of derivatives

### Derivatives of constant

c ′ = 0, где c = const### Derivatives of power functions

(x^{n})′ = n · x

^{n - 1}

### Derivatives of exponential functions

(a^{x})′ = a

^{x}· ln a

### Derivatives of exponential functions

(e^{x})′ = e

^{x}

### Derivatives of logarithmic functions

(log_{a} x)′ = | 1 |

x · ln a |

(ln x)′ = | 1 |

x |

### Derivative of Trigonometric Functions

(sin x)′ = cos x

(cos x)′ = -sin x

(tg x)′ = | 1 |

cos ^{2} x |

(ctg x)′ = - | 1 |

sin ^{2} x |

### Derivative of inverses trigonometric functions

(arcsin x)′ = | 1 |

√1 - x^{2} |

(arccos x)′ = - | 1 |

√1 - x^{2} |

(arctg x)′ = | 1 |

1 + x^{2} |

(arcctg x)′ = - | 1 |

1 + x^{2} |

### Derivative of Hyperbolic functions

(sh x)′ = ch x

(ch x)′ = sh x

(th x)′ = | 1 |

ch^{2} x |

(cth x)′ = - | 1 |

sh^{2} x |

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