Partial Derivative: Definition, Formulas and Examples | Partial Differentiation

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 = 𝛿f/𝛿x = 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 = 𝛿f/𝛿y = 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 = 𝛿f/𝛿z = 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. We will study this using different examples.

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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 = 𝛿f/𝛿x and f_y = 𝛿f/𝛿y

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^‘ = 𝛿²f/𝛿x² and f_y^‘ = 𝛿²f/𝛿y²

For example considered above example

f^‘_x = 𝛿(2xy)/𝛿x

f^‘_x(2,1) = 2.y = 2

f^‘_y = 𝛿(x²+6y)/𝛿y

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 = (𝛿f(x,y)/𝛿x).g(x,y) + f(x,y).(𝛿g(x,y)/𝛿x)

• u_y = (𝛿f(x,y)/𝛿y).g(x,y) + f(x,y).(𝛿g(x,y)/𝛿y)

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 = (𝛿f(x,y)/𝛿x).g(x,y) – f(x,y).(𝛿g(x,y)/𝛿x)/(g(x,y))²

• u_y = (𝛿f(x,y)/𝛿y).g(x,y) – f(x,y).(𝛿g(x,y)/𝛿y) /(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(𝛿f(x,y)/𝛿x)

• u_y = n. |f(x,y)|^n-1(𝛿f(x,y)/𝛿y)

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

𝛅z/𝛅t = 𝛅z/𝛅x. 𝛅x/𝛅t + 𝛅z/𝛅y. 𝛅y/𝛅t

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

𝛅z/𝛅u = 𝛅z/𝛅x. 𝛅x/𝛅u + 𝛅z/𝛅y. 𝛅y/𝛅u and 𝛅z/𝛅v = 𝛅z/𝛅x. 𝛅x/𝛅v + 𝛅z/𝛅y. 𝛅y/𝛅v

Total Derivative Vs Partial Derivative

Let us compare the total derivative and the partial derivative.

Parameter	Partial Derivative	Total Derivative
Denoted by	It is denoted by ∂	It is denoted by d
Definition	It measures how a function changes concerning one of its variables, holding all other variables constant.	It measures how a function changes concerning one of its variables, considering the changes in all other variables as well.
How it deals with other variables	It holds other variables constant.	It takes into account changes in all variables.
Example	For f(x, y) = x2y. The partial derivative wrt x is 2xy and wrt y is x2.	For f(x, y) = x2y. Total derivative is 2xy.dx/dt + x2y.dy/dt
Uses	Used in multivariable calculus, and optimization problems.	Used in differential equations to describe the whole system.

Partial Derivative of Natural Logarithm (In)

We will understand how can we find the partial derivative of the natural logarithm “In”. The steps remain the same except for the part that calculates the partial derivative of the function concerning one independent variable by considering all other variables as constant.

Example: Find partial derivative of F(x, y) = ln(xy)

F(x, y) = ln(xy)

Taking partial derivative with respect to x, we keep y constant

𝛅F/𝛅x = 1/xy.𝛅(xy)/𝛅x

𝛅F/𝛅x = (1/xy).y = 1/x

Taking the partial derivative with respect to y, we keep x constant

𝛅F/𝛅y = 1/xy.𝛅(xy)/𝛅y

𝛅F/𝛅y = 1/xy.x = 1/y

Applications of PartialDerivative

Various applications of partial derivative includes:

• Partial derivative is used in mathematical models that use complex equations. In such systems, the partial derivative helps to find the approximate solution of the system with efficiency.

• Partial derivatives are used in engineering-control systems. The control systems include dynamic systems that are complex to analyze with simple engineering tools. Therefore, partial derivates help to model and analyze such systems.

• Partial derivatives are used in chemical reactions to study the reaction kinetics of different reagents. They are mainly used to study reaction kinetics with changes in the concentration of different reagents and products.

• Partial derivatives are used in population dynamics to study how the population would change with changes in individual factors like birth rate, death rate, immigration, and calamities.

• Partial derivatives are used in the economics and finance sectors to predict future income. Modifications are done according to the results to maximize profits. For example, it helps to calculate a firm’s output in terms of input factors like labor or capital.

Partial Derivative Examples

Example 1: Find the partial differential coefficient of the function xy² with respect to y where x²+ xy + y²= 1.

Solution:

Let z = xy², we have to find the partial differential coefficient of z concerning y, that is, 𝛅z/𝛅y

We can write,

Let w = x²+ xy + y² = 1

Differentiating both sides concerning y, we get

𝛅w/𝛅y = 2xdx/dy + x + y.dx/dy+ 2y = 0

x + 2y = 0

f(x, y) = xy²

f(x,y) = (-2y).y²

f(x,y) = -2y³

𝛅f(x,y)/𝛅y = (-6).y²

Example 2: Find the partial differential coefficient of the function f(x,y,z) = x²y+ y²z+ xz with respect to x at x = 1, y = 2, z = 1.

Solution:

f(x,y,z) = x²y+ y²z+ xz

Calculating partial derivative with rewspect to x, we will consider y and z to be constants ∴

𝛅f(x,y,z)/𝛅x = 2xy + 0 + z

𝛅f(x,y,z)/𝛅x = 2xy +z

On putting the values x = 1, y = 2, z = 1

𝛅f(x,y,z)/𝛅x = 2.1.2 +1 = 5

Practice Problem

Here are some problems for practice purposes.

Question 1: Given, u = cos(x/y), x = e^t, y = t², find δu/δx at t = 1. Verify your result by direct substitution.

Question 2: Given, f(x, y) = e^xsin(y). Then evaluate δu/δy at x = 0 given x²+ y² = 1.

Question 3: Given, f(x, y) = ln(x/y).sin(y/x). Then evaluate δu/δy at x = 0 given x²+ y² = 1.

Question 4: Given, u = tan(x.y), x = sin(t), y = 3t², find δu/δx at t=1.Verify your result by direct substitution.

FAQs on Partial Derivative

What is Partial Derivative?

Partial derivative of a function is defined as derivative of any two variable function taking one variable as constant.

What is a partial derivative example?

Example of partial derivative includes, for function f(x, y) = 2x²y, its derivative with respect to y is, ∂f(x, y)/∂y = ∂{2x²y}/∂y = 2x².

What is ∂ called?

∂ in mathematics is symbol that denotes partial derivative.

What is partial derivative of Z?

Suppose Z is a function of x and y then, partial derivative of z with respect to x is calculated by differentiating z and taking y as constant.

Why is it called partial derivative?

Partaial derivative are named partial because, these derivative find derivative of any function treating other function as constant.

Why do we calculate an approximate solution and not the exact solution?

For complex equations like the ones given in the solved examples, finding an exact solution using the total derivative is an extremely complex task, especially for functions involving multiple variables. Therefore, we use the partial derivative to calculate the approximate solution.

What is the difference between partial derivatives and total derivatives?

Partial derivatives are the measure of change in a function concerning change in a single variable while taking all other variables as constant. However, the total derivative is the measure of change in the function concerning the change in all variables.

What is the partial derivative of a function z of variables x and y?

The partial derivative of z with respect to x is given by δz/δx. This will help to calculate the approximate change in z due to the change in x when y is constant.