Partial Derivative Formulas and Identities

A partial derivative is a derivative that is used in Differential Geometry and vector calculus. In context with mathematics, sometimes the function depends on more than two variables.

Since the function depends on where the variable in derivatives converts it into a partial derivative. The partial derivative is a very important topic in vector calculus. Suppose a function is  f(x,y); this suggests that this function is dependent on both the variables y and x

But, where y and x are interrelated or dependent on each other. The derivative of f is known as the partial derivative, where f is differentiated concerning x and y remains constant, and again f is differentiated concerning y and x remains constant. This article will discuss the formula of partial derivatives, rules for partial derivatives, and partial derivative examples. 

What are Partial Derivatives?

Suppose a function f(x,y) is dependent on two variables, y, and x, where y and x are independent and independent of each other. Then it is said that the function f is a partial function that depends on y and x. When f is differentiated between x and y, it stays constant, and again f is differentiated between y and x, it remains constant.

Symbol of Partial Derivative

In context with mathematics, the partial derivative of any mathematical function having different variables is its derivative for the one of that variable, where the remaining variables are held back as constants. The partial derivative of a mathematical function f for the differently x is severally denoted by f’x, fx, ∂xf, or ∂f/∂x. Where ∂ is the sign of the function or partial derivative.

Let’s take a look at an example:

Suppose there is a function f in y and x, then it will be represented as f(x, y). So, the partial function of the partial derivative of the function f with respect to x will be  ∂f/∂x, holding back y as a constant. ∂f/∂x is also called as the function or fx.

Rules of Formula of Partial Derivatives 

Like different derivatives, the rule of derivatives or partial derivatives also uses or follows some rules such as the quotient rule, chain rule, and product rule.

Product Rule:

The rule of derivatives of the partial derivative or the product rule of a partial derivative can be utilised for functions that are a product of different differentiable variables or functions. Such function can be written as mentioned below; 

In this rule, if:

u= f(x, y).g(x, y), then it can be described as:

ux  =  ∂u/∂x = g(x, y)  ∂f/∂x + f(x, y)  ∂g/∂x 

and, uy =  ∂u/∂y = g(x, y)  ∂f/∂y + f(x, y)  ∂g/∂y 

Quotient Rule:

In the Partial derivative, the quotient rule can be utilised when the function is made up of one differentiable function divided by another mathematical function; for such functions, they can be described as:

f(x, y) =   g(x, y)/h(x, y)

The partial derivative of y and x are mentioned below:

δf/δx = (h(x, y)δg/δx –  g(x, y)δh/δx)/h(x, y)2

δf/δy = (h(x, y)δg/δy –  g(x, y)δh/δy)/h(x, y)2

Chain Rule

In partial derivatives, the chain rule can be utilised for composite functions. The Composite function can be described as the thought of a function with the function. Let us take a differentiable function that is given as:

z = f(x, y)

And in this function, the variable is specified as:

x = g(t), and y = h(t)

A Chain rule can be described as:

dz/dt = (δf/δx)dx/dt + (δf/δy)dy/dt

Identities and Formulas of Partial Derivative 

Below-mentioned are some applications or identities for the function of partial derivative examples:

• If u= F(x, y) and both the variables y and x are  differentiable by t, y=h(t), and  x =g(t), Where we can consider this differentiation as a total differentiation

• The derivative of the total, partial derivative of u with respect to the t is df/dt = (∂f/∂x . dx/dt) + (∂f/∂y . dy/dt).

• If a mathematical function is defined as F(x), then x(u, v) can be described as:

 ∂f/∂u = ∂f/∂x. ∂x/∂u and ∂f/∂v = ∂f/∂x. ∂x/∂v

• If a mathematical function f(x, y), wherein the function  y(u, v) and x(u, v) are held back as constant, then it can be written as:

∂f/∂u = ∂f/∂x. ∂x/∂u + ∂f/∂y. ∂y/∂u

∂f/∂v = ∂f/∂x. ∂x/∂v + ∂f/∂y. ∂y/∂v

The Formula of Partial Derivatives

If the mathematical function For the formula of partial derivatives of f concerning x is denoted as ∂f/∂x and can be described as:

∂f/∂x = limδx→0 f(x + δx,y) – f(x,y)/δx

The solution for the partial derivative of f concerning Y then it can be denoted as ∂f/∂y and described as:

∂f/∂y = limδy→0 f(x,y + δy) – f(x,y)/δy

Conclusion

The partial derivative is a very important topic in vector calculus. A partial derivative is a derivative that is used in Differential Geometry and vector calculus. In context with mathematics, sometimes the function depends on more than two variables. Suppose a function f(x,y) is dependent on two variables, y, and x, where y and x are independent and independent of each other. Then it is said that the function f is a partial function that depends on y and x. When f is differentiated between x and y, it stays constant, and again f is differentiated between y and x.