Derivative Calculator

This calculator evaluates derivatives using analytical differentiation. It will also find local minimum and maximum, of the given function. The calculator will try to simplify result as much as possible.

There are examples of valid and invalid expressions at the bottom of the page.


$$ \begin{aligned} f(x) &= \sin(x) \\ \\\end{aligned} $$
$$ \begin{aligned} f~'(x) &= \cos(x) \\ \\\end{aligned} $$
$$ \begin{aligned} f~''(x) &= −sin(x) \\ \\\end{aligned} $$
$$ \begin{aligned} f~'''(x) &= −cos(x) \\ \\\end{aligned} $$



1. For powers use ^. Example: x12= x^12 ; ex+2= e^(x+2)
2. For square root use "sqrt". Example: x+1−−−−−√= sqrt(x+1).
3. Supported constants: e, pi
4. Supported functions: sqrt, ln ( use 'ln' instead of 'log'), e (use 'e' instead of 'exp')
Trigonometric functions: sin cos tan cot sec csc
Inverse trigonometric functions: acos asin atan acot asec acsc
Hyperbolic functions: sinh, cosh, tanh, coth, sech, csch

Examples of valid and invalid expressions

Function to integrate Correct syntax is Incorrect syntax is
$$ (2x+1)^6 $$ (2x+1)^6 [2x+1]^6
$$ \frac{10x + 1}{x^2-4} $$ (10x+1)/(x^2-4) 10x+1/x^2-4
$$ \left(ln(x)\right)^2 $$ ln(x)^2 ln^2(x)
$$ x ~ ln\left(\frac{x-1}{x+1}\right) $$ x*ln((x-1)/(x+1)) x*ln(x-1)/(x+1)

The derivative of a function f at a point x, written ƒ ′(x), is given by:

–                lim        ƒ(x + ∆x) — ƒ(x)
ƒ′(x) =  ∆s→0   ——————–
–                                       ∆x

if this limit exists.
the derivative of a function corresponds to the slope of its tangent line at one specific point. The following illustration allows us to visualise the tangent line (in blue) of a given function at two distinct points. Note that the slope of the tangent line varies from one point to the next. The value of the derivative of a function therefore depends on the point in which we decide to evaluate it. By abuse of language, we often speak of the slope of the function instead of the slope of its tangent line.
Derivatives of usual functions
Below you will find a list of the most important derivatives. Although these formulas can be formally proven, we will only state them here. We recommend you learn them by follows.
The constant function

Let ƒ(x) = k, where k is some real constant.
Then ƒ′(x) = (k)′ = 0

(8)′ = 0
(—5)′ = 0
(0,2321)′ = 0
The identity function †(x) = x
Let ƒ(x) = x , the identity function of x. Then
ƒ′(x) = (x)′ = 1

• The rule mentioned above applies to all types of exponents (natural, whole, fractional). It is however essential that this exponent is constant. Another rule will need to be studied for exponential functions (of type ax ).
• The identity function is a particular case of the functions of form
• x^n (with n = 1) and follows the same derivation rule : (x)′ = (x1)′ = 1 x1–1 = 1
x^0 = 1

It is often the case that a function satisfies this form but requires a bit of reformulation before proceeding to the derivative. It is the case of roots (square, cubic, etc.) representing fractional exponents.
An exponential function (of the form a^x with a Σ 0):

It is very easy to confuse the exponential function a^s with a function of the form x^n since both have exponents. They are, however, quite different. In an exponential function, the exponent is a variable.

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