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) : Functions

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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 (indefinite integral).

The derivation calculation (first order **derivative**) 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/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).

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**.

Use the primitive calculator tool available on dCode.

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