# Partial Derivative Calculator

Use this Partial Derivative Calculator below to solve your partial derivative related problems:

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What is a partial derivative?

How do you compute it? For what purposes is it used for?

If you’re looking for answers to questions like these, then you’ve come to the right page. We will discuss the above questions as well as the various areas where partial derivates are used.

### What is a Partial Derivative?

The term contains two words: “partial” and “derivative”. The derivative of any algebraic expression is calculated with respect to a certain specified variable. This is done by differentiating the given function or expression with respect to the specified variable and it symbolises the change in given function f(x) when the specified variable changes infinitesimally.

Also see our series on solving integral equations.

The derivative part is pretty clear when f(x) is composed of a single variable, but if it contains more than one variable then the inter-dependence of each variable also needs to be taken into account while calculating the derivative. And this is where the concept of “partial” derivative comes into play.

The partial derivative of a multi-variable expression with respect to a single variable is computed by differentiating the given function w.r.t. the desired variable whilst treating all other variables as constant, unlike the total differential where all variables can vary.

Partial derivatives symbolise the instantaneous change in a given function relative to the infinitesimal change of variable under consideration. It is extensively used in differential geometry and vector calculus. Also, for optimization purposes, partial derivatives play a major role in every field and the total derivative can be computed by step-by-step using partial derivatives.

## Examples & Usage of Partial Derivatives

As stated above, partial derivative has its use in various sciences, a few of which are listed here:

Optimization – Partial derivates are used for calculus-based optimization when there’s dependence on more than one variable.

Geometry – To compute rate at which a certain geometric quantity (volume, surface area, etc.) varies when a basic measurement (radius, height, length, etc) is varied.

Mathematical Physics – Partial derivates (rather partial differential equations) have extensive use in mathematical physics (and variational calculus, Fourier analysis, potential theory, vector analysis, etc).

Thermodynamics – Partial derivatives in this case can be thermal variables or ratios of some variables like mole fractions in the Gibbs energy equation.

Quantum Mechanics – Schrodinger wave equation and several other equations from quantum mechanics inherently use partial derivatives.

Economics – Most functions explaining economic behaviour statements like the behaviors depending on so and so variables in a particular manner, are obtained using the concept of partial derivates where independent variation in the behaviour is observed by varying the fundamental variables one by one.

In addition to these partial derivatives are used in many other areas of education to calculate the differentiation of a function partially with respect to a variable.

## Notations used in Partial Derivative Calculator

Let f be a function in x,y and z.

First order partial derivatives are represented by

$\dfrac{\partial f}{\partial x} = f_x$

Second order partial derivatives given by

$\dfrac{\partial^2 f}{\partial x^2} = f_{xx}$

The mix derivative is shown by

$\dfrac{\partial^2 f}{\partial x^2} = f_{xx}$

## Summarizing

As much use partial derivatives have, they are equally difficult to compute at higher levels and hence online partial derivative calculators are designed to help the users simplify their computations. Below we have presented one such calculator, equipped with the functions of computing partial derivatives to cater to all your computational needs.