Partial derivative is a mathematical concept used in vector calculus and differential geometry. If a function is dependent on two or more variables then its derivative is calculated in various types and one such type is partial derivative in which the derivative of any two or more variable functions is calculated taking one variable as constant.

Partial Derivative

In this article, we will learn about the partial derivative definition, formula, examples, examples and others in detail.

What is Partial Derivative?

A Partial derivative is calculated for a function f to approximate the value of the function concerning a certain parameter of the function. The term ‘partial’ is used to indicate that if the function is dependent on more than one variable, then the derivative will be taken considering one variable to calculate the change concerning that variable.

Example: For example, consider a function k which is an independent variable. If this function depends on two dependent variables l and m we can write k= f(l,m). Thus k is an independent variable which is represented in terms of dependent variables l and m. Now, we can calculate the partial derivative to see the approximation of the function due to the dependent variable.

Definition of the partial derivative of a function is,

Partial derivative of a function f dependent on x and y with respect to x is ∂f/∂x. Here ∂ is the symbol of the partial derivative.

Partial Derivative Symbol

Generally, ∂ this is the symbol of the partial derivative which is different from d.

Partial Derivative of a function is used to calculate the total change in the dependent variable due to the change in a particular independent variable keeping other variables constant

Partial Derivative Formula

Mathematically, consider a function f of dependent variables x, y, and z. Then the partial derivative of the function concerning x, y, and z can be written as

Partial derivative of a function with respect to x, keeping y and z constant:

f_x = ????f/????x = lim_h→0 (f(x+h, y, z) – f(x, y, z))/h

Partial derivative of a function with respect to y, keeping x and z constant:

f_y = ????f/????y = lim_h→0 (f(x, y+h, z) – f(x, y, z))/h

Partial derivative of a function with respect to z, keeping x and y constant:

f_z = ????f/????z = lim_h→0 (f(x, y, z+h) – f(x, y, z))/h

Partial Differentiation

Partial differentiation refers to the process of calculating the partial derivative of a given function. Mathematically Partial Differentiation gives the slope of tangent drawn to the graph of the function at any point.

Consider a function f(x,y) = x²y + 3y², we would like to see the first-order and second-order partial derivative of a function at x=2 and y=1.

Partial Derivatives of Different Orders

Depending on the order of derivative required, the partial derivatives can vary. Let us see how

First Order Partial Derivatives

Formula for calculating First Order Partial Derivatives is given by

f_x = ????f/????x and f_y = ????f/????y

For example considered above let’s calculate the value.

f(x,y) = x²y+3y²

f_x = 2xy + 0

f_x (2,1) = 4

f_y = x²+6y

f_y(2,1) = 4+6 = 10

Second Order Partial Derivatives

Formula for calculating second Order Partial Derivatives is given by

f_x^‘ = ????²f/????x² and f_y^‘ = ????²f/????y²

For example considered above example

f^‘_x = ????(2xy)/????x

f^‘_x(2,1) = 2.y = 2

f^‘_y = ????(x²+6y)/????y

f_y(2,1) = 6

Partial Derivative Rules

Let us see some rules used for calculating the partial derivative of a given function

Product Rule

This rule is used when a function is a product of two different functions i.e. u = f(x,y).g(x,y). According to the product rule, the partial derivative of this function will be,

• u_x = (????f(x,y)/????x).g(x,y) + f(x,y).(????g(x,y)/????x)

• u_y = (????f(x,y)/????y).g(x,y) + f(x,y).(????g(x,y)/????y)

Quotient Rule

This rule is used when a function is the quotient of two different functions i.e. u = f(x,y)/g(x,y). According to the quotient rule, the partial derivative of this function will be

• u_x = (????f(x,y)/????x).g(x,y) – f(x,y).(????g(x,y)/????x)/(g(x,y))²

• u_y = (????f(x,y)/????y).g(x,y) – f(x,y).(????g(x,y)/????y) /(g(x,y))²

Power Rule

This rule is used when a function is in the power of some number I.e u = (f(x,y))ⁿ. According to the power rule, the partial derivative of this function will be,

• u_x = n. |f(x,y)|^n-1(????f(x,y)/????x)

• u_y = n. |f(x,y)|^n-1(????f(x,y)/????y)

Chain Rule

The chain rule is a tool used for calculating the derivative of a multivariable function. According to the chain rule of derivatives

Chain Rule for One Independent Variable:

Let us consider two continuous functions u that are dependent on one variable t given by if x = g(t) andy=h(t). We have z = f(x, y) which is a differentiable function of x and y.

Then partial derivative

????z/????t = ????z/????x. ????x/????t + ????z/????y. ????y/????t

Chain Rule for Two Independent Variables:

Let us consider two continuous functions u that are dependent on two variables u and v given by x = g (u, v) and y = h (u, v). We have z = f(x, y) which is a differentiable function of x and y.We can write f as z = f (g (u, v), h (u, v)). Then partial derivative

????z/????u = ????z/????x. ????x/????u + ????z/????y. ????y/????u and ????z/????v = ????z/????x. ????x/????v + ????z/????y. ????y/????v

Total Derivative Vs Partial Derivative

Let us compare the total derivative and the partial derivative.