What is a Partial Derivative? Scope and Application

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What is a Partial Derivative

Differentiation is a common process in mathematics and physics. Differentiation is the process of finding the derivative of a function. This derivative is to compute the rate of change of something, and hence, Partial Derivative has numerous applications in the field of science and mathematics. Partial derivative can be handy to solve real-world problems.

Using derivatives, you can identify the rate of change of volume, area, speed, etc. Derivative usually refers to a function with a single driven variable, i.e., The function has a single dependent and single independent variable.

Learning and understanding derivatives can save you from multiple complex Mathematics problems. The more you understand, the better you become at it. Here, we will teach you some tips and tricks for elementary studying and competence.

So, the derivative can either be written as d(dependent variable) / d(independent variable). If y is dependent and x is independent, it can be written as:

dy/dx.

If x is dependent and Y is independent, it can be written as:

dx/dy.

A derivative of a function is usually computed using ‘The First Principle’ as shown below.

f’(a) = lim (f(a+h) – f(a)) /h

where the limit is h0.

Now, if you’re dealing with more complex functions or equations you might come across a term caked partial Derivative. What is this partial Derivative? Where to use it? Let’s find out.
Partial Derivative is the result of the differentiation of a function with several variables with respect to a single variable. To be precise, a function with many independent variables.

This makes the process of differentiation a bit more difficult. Both partial differentiation and differentiation comes under the topic calculus, which is the heart of mathematical physics.

Notation of Partial Derivative

The operator of the partial Derivative is denoted by ∂ and can be pronounced as ‘dou’. In some regions, it is also pronounced as ‘dough,’ but the former one is in use in most of the cases.

Now, a partial Derivative of a function containing variables a, b, c, d, and so on up to z concerning x is as follows:

∂a/∂x + ∂b/∂x + ….. + 1 +.. + ∂z/∂x

General representation is ∂f/∂x, where f is a function with many independent variables. The general notation found in most of the textbooks for ∂f/∂x is p and for ∂f/∂y is q. These notations need not be the same in every textbook.

Concept Of Derivatives

For becoming a pro at roots of calculus, you must have the basic knowledge and know
the concept of derivatives. It is the essence of Calculus.

    • It is defined as the ratio between the variation of two variables, that is, x and y.

      Slope = Change in y/Change in x

Partial Derivative Definition

Partial Derivatives are similar to limits. The First Principle is to apply for functions with many independent variables, and the partial Derivative is easier to obtain. Partial Derivative is similar to the actual derivative in many cases.

We can check differentiability at a given point using partial derivatives just as in the case of derivatives. If all the derivatives of a function ‘f’ at a point a exist in the neighborhood of the given point ‘a’ and are continuous at the same point, then the function can be said differentiable function.
However, continuity is still uncertain at a point by observing the partial derivatives. A function need not be continuous at a point ‘a’ even if all the partial derivatives of the function exists at the same point.

partial derivatives representation
partial derivatives representation

Therefore, you need to find out the derivative to determine continuity. However, we know that a function should be continuous to be drivable. Hence we can justify the continuity of a function in this way.
Double differentiation is also applicable for functions with multiple independent variables. This means that just as in the case of derivatives, there can be double partial derivatives. To depict double partial derivatives we use (∂^2) f/(∂x) ^2, and which sounds “dou Square f by dou x Square.”

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We can differentiate a partial derivative of x, i.e., ∂f/∂x  with respect y giving (∂^2) f/∂x∂y. which pronounces as “Dou square f by dou x dou y.” In many textbooks, representation is with the letter ‘s’ which need not be the same in every textbook.

Solving Partial Derivatives

Follow the same conventional way of normal derivatives to solve partial derivatives.  If you need to solve the function f= 5x + 6y where x and Y are functions of another variable ‘t,’ you can solve the function as follows:

∂f/∂x = 5+6×∂y/∂x
∂f/∂y = 5×∂x/∂y + 6

If you want to find the complete derivative or the total derivative of the function, you can solve the values and substitute in the below formula.

f’(x) = ∂f/∂x × dx/dx + ∂f/∂y × dy/dx

This formula gives you the final derivative of the function f with respect to x. If you want to find out the total derivative of the function f(x, y) with respect to y, you can multiply ∂f/∂x with dy/dx and ∂f/∂y with dy/dy which is one.

Applications of Partial Derivatives

Partial derivatives are an important part of every optimization problem with two or more variables. Since partial Derivatives operate on multiple variables, they extensively operate in the Firefox of mathematics and science.

Many Engineering applications employ the use of Partial derivatives. Using Partial derivatives is common in the field of economics when operating with multiple variables.
The image resizing software also employs the use of partial derivatives to estimate the energy of pixels.

Maxima and Minima using Partial Derivative
Maxima and Minima using Partial Derivative

Thermodynamics also has many equations with multiple variables which makes the partial derivatives key components in solving thermodynamic equations.

Since many equations in thermodynamics employ the use of partial derivatives, applications of the partial derivatives extend to the field of Astrophysics. With Partial derivatives, we can also find the relative maximums and minimums in mathematics. Moreover, we can compute absolute Maxima and absolute Minima using partial derivatives as well.

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