Derivative calculator allows steps by steps calculation of the derivative of a function with respect to a variable.

Derivative online

The **derivative calculator** allows to do symbolic **differentiation**
using the derivation property on one hand and the derivatives of the other usual functions.
**Derivative calculation** obtained is returned after being simplified, with calculation steps.

The **derivative calculator** may **calculate online the derivative** of any
**polynomial**.

For example, to **calculate online the derivative of the polynomial** following `x^3+3x+1`, just enter
derivative(`x^3+3x+1`),
after calculating result `3*x^2+3` is returned.

**Derivative calculator** is able to **calculate online all common derivatives**
: sin, cos, tan, ln, exp, sh, th, sqrt (square root) and many more ...

Thus, to obtain the derivative of the cosine function with respect to the variable x, you must enter derivative(`cos(x);x`), result `-sin(x)` is returned after calculation.

The derivative of a sum is the sum of the derivatives, calculator uses this property to differentiate functions.

To **calculate online the derivative of a sum**, simply type the mathematical expression that contains the sum,
specify the variable and apply derivative function.

For example, to calculate online the derivative of the sum of the following functions `cos(x)+sin(x)`, enter derivative(`cos(x)+sin(x);x`), after calculating result `cos(x)-sin(x)` is returned.

Note that **steps of the calculations** for obtaining the calculation of the derivative are also displayed by the function.

To **calculate online the derivative** of a **difference**, simply type the mathematical expression
that contains the difference, specify the variable and apply derivative function.

For example, to calculate online the derivative of the difference of the following functions `cos(x)-2x`, enter derivative(`cos(x)-2x;x`), after calculating result `-sin(x)-2` is returned.

It is noted that description and steps calculations of the derivative are also displayed by the function.

To **calculate online the derivative of a product**, just input the mathematical expression
that contains the product, specify the variable and apply derivative function.

For example, to calculate online the derivative of the product of the following functions `x^2*cos(x)`, enter derivative(`x^2*cos(x);x`), after calculating result `2*x*cos(x)-x^2*sin(x)` is returned.

Note that there also the derivative is calculated online with the description and the steps of the calculations.

To **calculate chain rule of derivatives**, just input the mathematical expression that contains
chain rule, specify the variable and apply derivative function.

For example, to calculate online the derivative of the chain rule of the following functions `cos(x^2)`, enter derivative(`cos(x^2);x`), after calculating result `-2*x*sin(x^2)` is returned.

Note that there also the derivative is calculated online with the description and the steps of the calculations.

To **differentiate a function**, it is necessary to know the following **calculation rules** and **formulas**:

- Formula for calculating the derivative of a function sum : (u+v)' = u'+v'
- Formula for calculating the derivative of a function product : (uv)' = u'v+uv'
- Formula for calculating the derivative of a function multiplied by a constant : (ku)' = ku'
- Formula for calculating the inverse derivative of a function : `(1/v)'` = `-(v')/v^2`
- Formula for calculating the derivative of the ratio of two functions : `(u/v)'` = `(u'v-uv')/v^2`

It is also necessary to know differentiated the usual functions which are in the following table :

f(x)= | f'(x)= |
---|---|

k | 0 |

x | 1 |

`x^n` | `n*x^(n-1)` |

`sqrt(x)` | `1/(2*sqrt(x))` |

g(ax+b) | `a*g'(ax+b)` |

abs(x) | `1` |

arccos(x) | `-1/sqrt(1-(x)^2)` |

arcsin(x) | `1/sqrt(1-(x)^2)` |

arctan(x) | `1/(1+(x)^2)` |

ch(x) | `sh(x)` |

cos(x) | `-sin(x)` |

cotan(x) | `-1/sin(x)^2` |

coth(x) | `-1/(sh(x))^2` |

exp(x) | `exp(x)` |

ln(x) | `1/(x)` |

log(x) | `1/(ln(10)*x)` |

`root(3)(x)` | `1/(3*(root(3)(x))^2)` |

sh(x) | `ch(x)` |

sin(x) | `cos(x)` |

sqrt(x) | `1/(2*sqrt(x))` |

tan(x) | `1/cos(x)^2` |

th(x) | `1/(ch(x))^2` |

By applying the derivation formulas and using the usual derivation table, it is possible to calculate any function derivative.
These are the calculation methods used by the **calc** to find the **derivatives**.

Derivative calculator allows steps by steps calculation of the derivative of a function with respect to a variable.

- derivative(function;variable), function is the function to differentiate
- It is also possible to use the Leibniz notation, using the symbol `d/dx`

To calculate the derivative of the function sin(x)+x with respect to x, you must enter :

- derivative(`sin(x)+x;x`) or
- derivative(`sin(x)+x`) , when there is no ambiguity concerning the variable.

The function will return 1+cos(x).

Calculate online with derivative (derivative calculator)
See also :

- Antiderivative calculator : antiderivative. The antiderivative calculator allows to calculate a primitive online with detail and calculation steps.
- Derivative calculator : derivative. Derivative calculator allows steps by steps calculation of the derivative of a function with respect to a variable.
- Integral calculator : integral. The integral function calculates online the integral of a function between two values.
- Even or odd function calculator : is_odd_or_even_function. Calculator for determining whether a function is an even function and an odd function.
- Limit of a function : limit. The limit calculator allows the calculation of the limit of a function with the detail and the calculation steps.
- Partial fraction decomposition : partial_fraction_decomposition. The calculator allows a rational fraction to be broken down into simple elements.
- Calculate the Taylor expansion of a function : taylor_series_expansion. The taylor series calculator allows to calculate the Taylor expansion of a function.