Tool for calculating function derivatives (simple derivative or partial derivative). Formal calculator from an expression f(x) of the function to be differentiated.

Derivative - dCode

Tag(s) : Functions

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Use the derivative calculator above by indicating a single variable (that of the partial derivative).

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

Simply put, the derivative of a function is the measure of the change in the function at a given point, it tells how fast the function is changing at that point.

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 (or first derivative) calculation applies the general formula $$ \frac{d}{dx}f = f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} $$

In practice, this limit calculation is sometimes laborious, it is easier to learn the 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/differentiate in order 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.

A partial derivative is a derivative that only applies to one variable, leaving the others intact.

On dCode, indicate a single variable if the function has several to obtain a partial derivative.

The more useful derivatives are:

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

constant/number | $$ k \in \mathbb{R} $$ | $$ 0 $$ |

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

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

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

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

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

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

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

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

sine | $$ \sin(x) $$ | $$ \cos(x) $$ |

cosine | $$ \cos(x) $$ | $$ - \sin(x) $$ |

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

secant | $$ \sec(x) = \frac{1}{\cos(x)} $$ | $$ \frac{\tan(x)}{\cos(x)} \\ = \sec(x)\tan(x) \\ = \frac{2\sin(x)}{1+\cos(2x)} $$ |

cosecant | $$ \csc(x) = \frac{1}{\sin(x)} $$ | $$ -\frac{\cos(x)}{\sin^2(x)} \\ = -\cot(x)\csc(x) \\ = \frac{2\cos(x)}{-1+\cos(2x)} $$ |

cotangent | $$ \cot(x) = \frac{1}{\tan(x)} $$ | $$ - \frac{1}{\sin^2(x)} \\ = -1-\cot^2(x) \\ = -\csc^2(x) \\ = \frac{2}{-1+\cos(2x)} $$ |

arcsine | $$ \arcsin(x) $$ | $$ \frac{1}{\sqrt{1-x^2}} $$ |

arccosine | $$ \arccos(x) $$ | $$ -\frac{1}{\sqrt{1-x^2}} $$ |

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

sine hyperbolic | $$ \sinh(x) $$ | $$ \cosh(x) $$ |

cosine hyperbolic | $$ \cosh(x) $$ | $$ \sinh(x) $$ |

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

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

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

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

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

Common derivatives of compound functions to know are:

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

compound function | $$ g \circ f $$ | $$ (g' \circ f)\times f' $$ |

function power n (exponentiation) | $$ f^n $$ | $$ n f^{n - 1} f' $$ |

sine of function | $$ \sin(f) $$ | $$ f' \cos(f) $$ |

cosine of function | $$ \cos(f) $$ | $$ - f' \sin(f) $$ |

exponential of function | $$ \exp(f) $$ | $$ f' \exp(f) $$ |

square root of function (positive function) | $$ \sqrt{f} $$ | $$ \frac{f'}{2\sqrt{f}} $$ |

logarithm of function (positive function) | $$ \ln(f) $$ | $$ \frac{f'}{f} $$ |

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).

The derivative calculation is the inverse operation of the primitive/antiderivative calculation (indefinite integral).

dCode has a tool for calculating primitives.

A derivator is a mathematical operator that has nothing to do with the derivative operation.

It is nevertheless possible, by extension, to call the dCode tool on this page an online *derivator* allowing to calculate derivatives.

dCode retains ownership of the "Derivative" source code. Except explicit open source licence (indicated Creative Commons / free), the "Derivative" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Derivative" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Derivative" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

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Cite as source (bibliography):

*Derivative* on dCode.fr [online website], retrieved on 2023-12-06,

- Derivative Calculator
- Partial Derivative Calculator
- What is a derivative? (Definition)
- How to calculate a derivative?
- How to calculate a partial derivative?
- What is the list of common derivatives?
- What is the list of compound function derivatives?
- How to calculate a second derivative?
- What is the difference between derivative and primitive?
- What is a derivator?

derivative,function,differentiation,calculator,integral,velocity,acceleration,tangent

https://www.dcode.fr/derivative

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