Online Derivative Calculator

More than just an online derivative solver

Wolfram|Alpha is a great calculator for first, second, and third derivatives; derivatives at a point; and partial derivatives. Learn what derivatives are and how Wolfram|Alpha calculates them.

Tips for entering queries

Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for a derivative.

`d/dx (sin^-1(x))`

derivative of arcsin

`d/dx (log(x))`

derivative of lnx

`d/dx (sec^2(x))`

derivative of sec^2

`d^2/dx^2 (sin^2 (x))`

second derivative of sin^2

`\partial/(\partialx)|_(x=0) (tan^-1 (x))`

derivative of arctanx at x=0

`\partial/(\partial x) ((x^2+y)/(x+y^2))`

differentiate (x^2+y)/(y^2+x) with respect to x

What are derivatives?

The derivative is an important tool in calculus that represents an infinitesimal change in a function with respect to one of its variables.

Given a function `f(x)`, there are many ways to denote the derivative of `f` with respect to `x`. The most common ways are `(df)/dx` and `f'(x)`. When a derivative is taken `n` times, the notation `(d^n f)/(dx^n)` or `f^n(x)` is used. These are called higher order derivatives. Note for second order derivatives, the notation `f''(x)` is often used.

At a point `x = a`, the derivative is defined to be `f'(a) = lim_(h->0)(f(a+h)-f(a))/(h)`. This limit is not guaranteed to exist, but if it does, `f(x)` is said to be differentiable at `x = a`. Geometrically speaking, `f'(a)` is the slope of the tangent line of `f(x)` at `x = a`.

As an example, if `f(x) = x^3`, then

`f'(x) = lim_(h->0)((x + h)^3 - x^3)/(h) = 3 x^2`,

and then we can compute `f''(x)`:

`f''(x) = lim_(h->0)(3(x + h)^2 - 3x^2)/(h) = 6 x`.

The derivative is a powerful tool with many applications. For example, it is used to find local/global extrema, find inflection points, solve optimization problems, and describe the motion of objects.

Learn more »

How Wolfram|Alpha calculates derivatives

Wolfram|Alpha calls Mathematica's `D` function, which uses a table of identities much larger than one would find in a standard calculus textbook. It uses "well known" rules such as the linearity of the derivative, product rule, power rule, chain rule, so on. Additionally, `D` uses "lesser known" rules to calculate the derivative of a wide array of special functions. For higher order derivatives, certain rules like the general Leibniz product rule can speed up calculations.