Tool to compute derivates. The differentiation is a fundamental tool when analysing a function, it allows to measure the sensitivity to change of a function.

Derivative - dCode

Tag(s) : Mathematics

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Tool to compute derivates. The differentiation is a fundamental tool when analysing a function, it allows to measure the sensitivity to change of a function.

Mathematicians have defined derivatives using the formula $$ \frac{d}{dx}f = f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} $$ The derivative calculation is the inverse operation of primitive calculation. dCode knows all derivatives, one has only to indicate the function and the variable to derivate with.

The more useful derivatives are:

Name | Function | Derivative |
---|---|---|

constant | $$ k, in, \mathbb{R} $$ | $$ 0 $$ |

variable | $$ x $$ | $$ 1 $$ |

power n | $$ x^n $$ | $$ n x^{n-1} $$ |

negative power | $$ x^{-n} $$ | $$ -n x^{-n-1} $$ |

fraction | $$ \frac{1}{x} $$ | $$ -\frac{1}{x^2} $$ |

inverse power | $$ \frac1{x^n} $$ | $$ -\frac n{x^{n+1}} $$ |

root | $$ \sqrt{x} $$ | $$ -n x^{-n-1} $$ |

nth root | $$ \sqrt[n]x $$ | $$ \frac1{n\sqrt[n]{x^{n-1}}} $$ |

fractional power | $$ x^{1/n} $$ | $$ (1/n)x^{(1/n)-1} $$ |

neperian logarithm | $$ \ln |x| $$ | $$ \frac{1}{x} $$ |

logarithm base a | $$ \log_a |x| $$ | $$ \frac{1}{x \ln a} $$ |

exponential | $$ e^x $$ | $$ e^x $$ |

exponent | $$ a^x $$ | $$ a^x \ln a $$ |

sinus | $$ \sin x $$ | $$ \cos x $$ |

cosinus | $$ \cos x $$ | $$ - \sin x $$ |

tangent | $$ \tan x $$ | $$ \frac{1}{\cos^2 x} = 1+\tan^2 x $$ |

cotangent | $$ \cot x $$ | $$ - \frac{1}{\sin^2 x} = -1-\cot^2 x $$ |

arcsinus | $$ \arcsin x $$ | $$ \frac{1}{\sqrt{1-x^2}} $$ |

arccosinus | $$ \arccos x $$ | $$ -\frac{1}{\sqrt{1-x^2}} $$ |

arctangent | $$ \arctan x $$ | $$ \frac{1}{1+x^2} $$ |

sinus hyperbolic | $$ \sinh x $$ | $$ \cosh x $$ |

cosinus hyperbolic | $$ \cosh x $$ | $$ \sinh x $$ |

tangent hyperbolic | $$ \tanh x $$ | $$ \frac{1}{\cosh^2 x} = 1 - \tanh^2 x $$ |

cotangent | $$ \coth $$ | $$ \frac{-1}{\sinh^2 x} = 1 - \coth^2 x $$ |

arcsinus hyperbolic | $$ \arcsinh x $$ | $$ \frac{1}{\sqrt{1+x^2}} $$ |

arccosinus hyperbolic | $$ \arccosh x $$ | $$ \frac{1}{\sqrt{x^2-1}} $$ |

arctangent hyperbolic | $$ \arctanh x $$ | $$ \frac{1}{1-x^2} $$ |

It is equivalient to derivate twice, for dCode, indicate twice the same variable.

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Source : http://www.dcode.fr/derivative

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