The **partial derivative** is used in vector calculus and differential geometry. In Mathematics, sometimes the function depends on two or more variables. Here, the derivative converts into the partial derivative since the function depends on several variables. In this article, We will learn about the definition of partial derivatives, its formulas, partial derivative rules such as chain rule, product rule, quotient rule with more solved examples.

**Table of contents:**

## Partial Derivative Definition

Suppose, we have a function f(x,y), which depends on two variables x and y, where x and y are independent of each other. Then we say that the function f partially depends on x and y. Now, if we calculate the derivative of f, then that derivative is known as the partial derivative of f. If we differentiate function f with respect to x, then take y as a constant and if we differentiate f with respect to y, then take x as a constant.

## Partial Derivative Symbol

In mathematics, the partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant.Â The partial derivative of a function f with respect to the differently x is variously denoted by fâ€™_{x},f_{x}, âˆ‚_{x}f or âˆ‚f/âˆ‚x. Here âˆ‚ is the symbol of the partial derivative.

**Example**: Suppose f is a function in x and y then it will be expressed by f(x,y). So, the partial derivative of f with respect to x will be âˆ‚f/âˆ‚x keeping y as constant. It should be noted that it is âˆ‚x, not dx.Â âˆ‚f/âˆ‚x is also known as f_{x}.

## Partial Derivative Formula

If f(x,y) is a function, where f partially depends on x and y and if we differentiate f with respect to x and y then the derivatives are called the partial derivative of f. The formula for partial derivative of f with respect to x taking y as a constant is given by;

## Partial Derivative Rules

Same as ordinary derivatives, partial derivatives follow some rule like product rule, quotient rule, chain rule etc.

### Product Rule

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

### Quotient Rule

If u = f(x,y)/g(x,y), where g(x,y) â‰ 0, then;

### Power Rule

If u = [f(x,y)]^{n} then, the partial derivative of u with respect to x and y defined as;

u_{x }= n|f(x,y)|^{n-1}âˆ‚f/âˆ‚x

And u_{y} = n|f(x,y)|^{n-1}âˆ‚f/âˆ‚y

### Chain Rule

Here, the chain rule for one independent variable and two independent variables are given below:

**Chain Rule for One Independent variable:**

Consider that, if x = g(t) andÂ y=h(t) are the differentiable functions of t, andÂ z = f(x, y) which is a differentiable funtion of x and y. Thus z can be written as z = f(g(t), h(t)), is a differentiable function of t, then the partial derivative of the function with respect to the variable “t” is given as:

\(\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}.\frac{\partial x}{\partial t} +\frac{\partial z}{\partial y}.\frac{\partial y}{\partial t}\)Here, the ordinary derivatives are determined at “t”, whereas the partial derivatives are evaluated at (x, y)

**Chain Rule for Two Independent variables:**

Assume that x = g (u, v) and y = h (u, v) are the differentiable functions of the two variables u and v, and also z = f (x, y) is a differentiable function of x and y, then z can be defined as z = f (g (u, v), hÂ (u, v)), which is a differentiable function of u and v. Thus, the partial derivative of the function with respect to the variables are given as:

\(\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} +\frac{\partial z}{\partial y}\frac{\partial y}{\partial u}\)and

\(\frac{\partial z}{\partial v} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial v} +\frac{\partial z}{\partial y}\frac{\partial y}{\partial v}\)## Partial Derivative of Natural Logarithm (In)

To find the partial derivative of natural logarithm “In”, we have to proceed with the same procedure as finding the derivative of the normal function. But, here when we calculate the partial derivative of the function with respect to one independent variable taking other as constant and follow the same thing with others.

**Also, see:**

## Partial Derivative Examples

**Example 1:Â Determine the partial derivative of the function: f (x,y) = 3x + 4y.**

**Solution:**

Given function:Â f (x,y) = 3x + 4y

To findÂ âˆ‚f/âˆ‚x, keep y as constant and differentiate the function:

Therefore,Â âˆ‚f/âˆ‚x = 3

Similarly, to findÂ Â âˆ‚f/âˆ‚y,Â keep x as constant and differentiate the function:

Therefore,Â âˆ‚f/âˆ‚y = 4

**Example 2:Â Find the partial derivative of f(x,y) = x ^{2}y + sin x + cos y.**

**Solution:**

Now, find out f_{x }first keeping y as constant

f_{x }= âˆ‚f/âˆ‚x = (2x) y + cos x + 0

= 2xy + cos x

When we keep y as constant cos y becomes a constant so its derivative becomes zero.

Similarly, finding f_{y}

f_{y }= âˆ‚f/âˆ‚y = x^{2} + 0 + (-sin y)

= x^{2} – sin y

**Example 3:Â FindÂ Â âˆ‚f/âˆ‚x,Â Â âˆ‚f/âˆ‚y,Â Â âˆ‚f/âˆ‚z for the given function, f(x, y, z) = x cos z + x ^{2}y^{3}e^{z}**

**Solution:**

To findÂ Â âˆ‚f/âˆ‚x,Â Â âˆ‚f/âˆ‚y,Â Â âˆ‚f/âˆ‚z

Given Function:Â f(x, y, z) = x cos z + x^{2}y^{3}e^{z}

âˆ‚f/âˆ‚x =Â cos z + 2xy^{3}e^{z}

âˆ‚f/âˆ‚y =Â 3x^{2}y^{2}e^{z}

âˆ‚f/âˆ‚z =-x sin z + x^{2}y^{3}e^{z}

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