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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