Partial Derivative of Functions

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What is Partial Derivative of Functions?

The partial derivative of functions is one of the most important topics in calculus. Let's say we have a function f(x,y); this implies that this is a function that depends on both the variables x and y where x and y are not dependent on each other. In this type of function, we can assume that function f partially depends on x and partially on y. The derivative of f is known as the partial derivative of f. When f is differentiated with respect to x, y remains constant, and when f is differentiated with respect to y, here x remains constant.


For example:

Suppose f is a function in x and y then it will be denoted by f(x,y) So the partial derivative of f with respect to x will be ∂f/მx considering that y is constant.


Suppose f(x,y) is a function where f partially depends on x and partially on y. Here if f is differentiated with respect to x and y, then the derivatives are known as the partial derivative of f with respect to x and y, which can also be called a partial derivative of x y.


The formula for partial derivative of f with respect to x, considering y as constant is:


Fx = ∂f/∂x = limh→0 f(x+ h,y) - f(x,y)/h


The formula for partial derivative of f with respect to y, considering x as constant is :


Fy = ∂f/∂y = limh→0 f(x,y+h) - f(x,y)/h


\[\frac{\partial f}{\partial x}  = \lim_{h \rightarrow 0} \frac{f(x + h, y) - f(x, y)}{h}\]


\[\frac{\partial f}{\partial y}  = \lim_{k \rightarrow 0} \frac{f(x, y + k) - f(x, y)}{h}\]


Partial Derivative Formulas and Identities

Keeping in mind the definition of functions, there are some listed identities for partial derivatives.

  1. If U = f(x,y) and both the variables x and y are differentiable of t i.e. x = g(t) and y = h(t), here we can consider differentiation as total differentiation.

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

  3. If f is a function that is defined as f(x), where x(u,v) then

∂f/∂u = ∂f/∂x . ∂x/∂u and

∂f/∂v = ∂f/∂x . ∂x/∂v

Suppose f = f(x,y) and y is a implicit function it means y itself is a function x, then

df/dx = ∂f/∂x + ∂f/∂y . dy/dx

  1. If f(x,y), where x(u,v) and y(u,v) is constant, then

∂f/∂u = ∂f/∂x . ∂x/∂u + ∂f/∂y . ∂y/∂u  and

∂f/∂v = ∂f/∂x . ∂x/∂v + ∂f/∂y . ∂y/∂v


First Partial Derivative

If U = f(x,y) then the partial derivative of f with respect to x is defined as ∂f/∂x and denoted by

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


And, the partial derivative of f with respect to y is defined as ∂f/∂y and is denoted by

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


These are known as the partial derivatives of f or partial derivative x y. We calculate the partial derivative of f with respect to x, considering y as constant and vice versa.


Double Partial Derivative

When you differentiate the first partial derivative, you will be able to find out the second-order partial derivative. The second-order derivative is also known as a double partial derivative. The partial differentiation fxy and fyx are distinguished by the order on which f is successively differentiated with respect to x and y. The two partial derivatives don't need to be equal.


Mixed Partial Derivative

In a scenario where the second-order partial derivative exists, you will be able to find out the mixed partial derivative or cross partial derivative. The terms fxy and fyx are known as the mixed partial derivatives of f or mixed partial derivatives x y.


High-Order Partial Derivative

Second and higher-order partial derivatives are defined in comparison to the higher-order derivatives of univariate functions. For the function f (x, y, . .) the "own" second partial derivative with respect to x with y as a constant is simply the partial derivative of the partial derivative.


Solved Example

1. Consider the function f(x,y) = 5x4y2 + 6x2y3 . Find fx and fy.


Solution

Given f(x,y) = 5x4y2 + 6x2y3

Take y as constant and differentiate the given function w.r.t x to find fx

fx = 20x3y2+12xy3

Take x as constant and differentiate the given function w.r.t y to find fy

fy = 5x4(2y)+6x2(3y2) = 10x4y+18x2

FAQ (Frequently Asked Questions)

1. What is a Partial Derivative of a Function? A Small Brief on its Symbol.

Answer: In mathematics, the partial derivative of a function of several variables is defined as the derivative of the function with respect to one of those variables with consideration of all the other variables as constant. The partial derivative x y is mainly used in vector calculus and differential geometry. The symbol used for identifying partial derivatives is ∂. This symbol was known to be first used by Marquis de Condorcet in the year 1770, and he used it for partial differences. Adrien-Marie Legendre created the modern partial derivative notation in the year 1786, but he later abandoned it. Later it was introduced by Carl Gustav Jacob Jacobi in the year 1841.

2. What is the Difference Between Derivative and Partial Derivative?

Answer: The derivative is applied to those functions that have only one independent variable, whereas the partial derivative is applied to those functions that consist of more than one independent variable. In partial derivative, the function is differentiated with respect to one variable considering the other variables constant, whereas in derivatives, the function is differentiated with respect to all the variables with no constant. In the case of partial derivative x y, f is differentiated with respect to x, considering y as constant or vice versa. In the case of derivative x y, f is differentiated with respect to both the variables.