# Partial Differentiation

Partial differentiation is used to differentiate mathematical functions having more than one variable in them. In ordinary differentiation, we find derivative with respect to one variable only, as function contains only one variable. So partial differentiation is more general than ordinary differentiation. Partial differentiation is used for finding maxima and minima in optimization problems. Let us discuss it in details.

## Partial Differentiation

Itâ€™s another name is Partial Derivative. It is a derivative where we hold some independent variable as constant and find derivative with respect to another independent variable.

For example, suppose we have an equation of a curve with X and Y coordinates in it as 2 independent variables. To find the slope in the direction of X, while keeping Y fixed, we will find the partial derivative. Similarly, we can find the slope in the direction of Y (keeping X as Fixed).

Partial differentiation builds with the use of concepts of ordinary differentiation. So we should be familiar with the methods of doing ordinary first-order differentiation.

Obviously, for a function of one variable, its partial derivative is the same as the ordinary derivative.Â  Constants, when added to functions, will differentiate to 0 but constants multiplying the functions, are retained and so still multiply the derivative.

### Mathematical Representation

HereÂ fâ€™xÂ to mean “the partial derivative with respect to x”. Its another very common notation is to use a backward d (âˆ‚) like this: âˆ‚fâˆ‚xÂ = 2x

This is the same as: Â fâ€™xÂ = 2x âˆ‚ is also called “del” or “dee” or “curly dee”

#### Example with Explanation

Take a function of one variable x:

f(x) = x2

It’s derivative using power rule:

First order derivative ::Â Â Â Â  fâ€™(x) = 2x

Now take a function of two variablesÂ x and y:

f(x,y) = x2Â + y3

To find itsÂ partialÂ derivativeÂ with respect to xÂ we considerÂ y as a constant:

Partial derivative wrt X ::Â Â  fâ€™xÂ = 2x + 0

= 2x

Now, to find the partial derivativeÂ with respect to y, we considerÂ x as a constant:

Partial derivative wrt Y ::Â  fâ€™yÂ = 0 + 3y2

= 3y2

For more than two variables we will consider all other variables as if they are constants.

Source: freepik.com

#### Practical Implication

Consider a Cylindrical object. The equation to find volume is:

V =Â Ï€Â r2Â h

Also, We can write that in multi-variable form asÂ Â Â  Â f(r,h) =Â Ï€Â r2Â h

For the partial derivative with respect to r we holdÂ h constant, and rÂ changes: fâ€™rÂ =Â Ï€Â (2r) h = 2Ï€rh

Here derivative of r2Â with respect to r is 2r, andÂ Ï€ is a constant and we assume h as constant. It says that, as only the radius changes (by the negligible amount), the volume changes by 2Ï€rh. It is like that we add another skin with a circle’s circumference 2Ï€r and a height of h.

For the partial derivative with respect to h, we holdÂ r as a constant:

fâ€™hÂ =Â Ï€Â r2Â (1)

=Â Ï€r2

Ï€Â and r2Â both are constants, and the derivative of h with respect to h is 1.

It says that as only the height changes (by the negligible amount, the volume changes byÂ Ï€r2.

It is like we add the thinnest disk with negligible height on top with a circle’s area ofÂ Ï€r2.

## Solved Example onÂ Partial Differentiation

Question-1: Find the partial derivative of the following function (in x and y) with respect to x and y separately. f(x,y) = 2x2Â + 4xy

Answer:Â  With respect to X : fâ€™xÂ = 4x + 4y

With respect to Y : fâ€™yÂ = 0 + 4x = 4x

Question-2 :Â Find theÂ partial derivativesÂ ofÂ function g given as:

g(x,y,z) = x4Â âˆ’ 3xyz.

Answer:Â  Function isÂ  g(x,y,z) = x4Â âˆ’ 3xyz

Wrt xÂ  :: âˆ‚gâˆ‚xÂ = 4x3Â âˆ’ 3yz

Wrt y :: âˆ‚gâˆ‚yÂ = âˆ’3xz

Therefore Wrt z :: âˆ‚gâˆ‚zÂ = âˆ’3xy

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