# 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 |

Factoring: Some special cases
Formulas and properties of exponents
Formulas and properties of nth root
Formulas and properties of logarithms
Formulas and properties of arithmetic sequence
Formulas and properties of geometrical sequence
Trigonometry formulasDerivative formulas
Integrals table

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