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# What is partial differentiation?

## Overview

We use partial differentiation in vector calculus and differential geometry.

When the function depends on two or more variables then the derivative changes into partial derivative.

A partial derivative of a function of several variables is an ordinary derivative with respect to one of the variables when all the other variables remain constant.

We use all the rules of differentiation but the condition is that if we are differentiating partially with respect to one variable, all the other variables are treated as constants.

## Definition of Partial Differentiation

Suppose we have a function f(x, y), which is dependent on two variables x and y. x and y are independent of each other.

Then we will say that the function ‘f’ partially depends on these two variables.

The derivative of the function ‘f’ is called the partial derivative of ‘f’.

## Notations

Suppose we have a function u = f (x, y) of two independent variables x and y. If we keep y constant and x varies then function u becomes a function of x only. The derivative of u with respect to x, keeping y as constant is called partial derivative of ‘u’, with respect to ‘x’.

The following notations we use for partial differentiation-

Similarly, the partial derivative of ‘u’ with respect to ‘y’ keeping x as constant is denoted by

Product and quotient rules of partial derivative

Product rule:

If u = f(x, y).g(x, y), then

And

Quotient rule:

If u = f(x, y)/g(x, y), then [note- g(x, y) should be non-zero]

And

Partial derivative of higher orders

Let u = f(x, y) then and will be the function of x and y, and these functions can again be differentiated partially with respect to x and y respectively.

These are called the second-order partial derivatives.

Note-

Solved examples

Example: Differentiate the following function partially with respect to x and y.

Solution:

Here the function is-

Now differentiating partially with respect to ‘x’, we get-

Similarly differentiating partially with respect to ‘y’, we get-

Example: Find the value of of the following function-

Solution:

Here we have-

First we will partially differentiate the given function with respect to ‘z’

Now

Finally

Which can be written as-

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