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

Mathematicians have defined derivatives by the formula $$ \frac{d}{dx}f = f\'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} $$

The derivative of a function $ f $ is denoted $ f' $ (with an apostrophe named prime) or $ \frac{d}{dx}f $ where $ d $ is the derivative operator and $ x $ the variable on which to derivate.

The derivative calculation is the inverse operation of primitive calculation.

The derivation calculation is based mainly on a list of usual derivatives, already calculated and known (see below).

On dCode, the derivative calculator knows all the derivatives, indicate the function and the variables on which to derivate to obtain the result of the derivative computation.

Example: $$ f(x) = x^2+\sin(x) \Rightarrow f'(x) = 2 x+\cos(x) $$

Derivative calculus is often used in physics to compute a velocity.

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 | $$ \sqrt{x} $$ | $$ \frac 1{2\sqrt{x}} $$ |

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

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

natural 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 | $$ \mathrm{arcsinh} x $$ | $$ \frac{1}{\sqrt{1+x^2}} $$ |

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

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

A second derivative is equivalent to compute the derivative twice, for dCode, indicate twice the same variable to get the second derivative.

Second derivative calculation is often used in physics to compute acceleration (derivative of velocity).

Use the primitive calculator tool available on dCode.

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

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