# 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.

## 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.

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