Partial Derivative

The partial derivative is used in vector calculus and differential geometry. In Mathematics, sometimes the function depends on two or more variables. Here, the derivative converts into the partial derivative since the function depends on several variables. In this article, We will learn about the definition of partial derivatives, its formulas, partial derivative rules such as chain rule, product rule, quotient rule with more solved examples.

Table of contents:

Partial Derivative Definition

Suppose, we have a function f(x,y), which depends on two variables x and y, where x and y are independent of each other. Then we say that the function f partially depends on x and y. Now, if we calculate the derivative of f, then that derivative is known as the partial derivative of f. If we differentiate function f with respect to x, then take y as a constant and if we differentiate f with respect to y, then take x as a constant.

Partial Derivative Symbol

In mathematics, the partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant. The partial derivative of a function f with respect to the differently x is variously denoted by f’x,fx, ∂xf or ∂f/∂x. Here ∂ is the symbol of the partial derivative.

Example: Suppose f is a function in x and y then it will be expressed by f(x,y). So, the partial derivative of f with respect to x will be ∂f/∂x keeping y as constant. It should be noted that it is ∂x, not dx. ∂f/∂x is also known as fx.

Partial Derivative Formula

If f(x,y) is a function, where f partially depends on x and y and if we differentiate f with respect to x and y then the derivatives are called the partial derivative of f. The formula for partial derivative of f with respect to x taking y as a constant is given by;

Partial Derivative Formula

Partial Derivative Rules

Same as ordinary derivatives, partial derivatives follow some rule like product rule, quotient rule, chain rule etc.

Product Rule

If u = f(x,y).g(x,y), then,

Partial Derivative-Product Rule

Quotient Rule

If u = f(x,y)/g(x,y), where g(x,y) ≠ 0, then;

Partial Derivative-Quotient Rule

Power Rule

If u = [f(x,y)]n then, the partial derivative of u with respect to x and y defined as;

ux = n|f(x,y)|n-1∂f/∂x

And uy = n|f(x,y)|n-1∂f/∂y

Chain Rule

Here, the chain rule for one independent variable and two independent variables are given below:

Chain Rule for One Independent variable:

Consider that, if x = g(t) and  y=h(t) are the differentiable functions of t, and  z = f(x, y) which is a differentiable funtion of x and y. Thus z can be written as z = f(g(t), h(t)), is a differentiable function of t, then the partial derivative of the function with respect to the variable “t” is given as:

\(\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}.\frac{\partial x}{\partial t} +\frac{\partial z}{\partial y}.\frac{\partial y}{\partial t}\)

Here, the ordinary derivatives are determined at “t”, whereas the partial derivatives are evaluated at (x, y)

Chain Rule for Two Independent variables:

Assume that x = g (u, v) and y = h (u, v) are the differentiable functions of the two variables u and v, and also z = f (x, y) is a differentiable function of x and y, then z can be defined as z = f (g (u, v), h (u, v)), which is a differentiable function of u and v. Thus, the partial derivative of the function with respect to the variables are given as:

\(\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} +\frac{\partial z}{\partial y}\frac{\partial y}{\partial u}\)

and

\(\frac{\partial z}{\partial v} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial v} +\frac{\partial z}{\partial y}\frac{\partial y}{\partial v}\)

Partial Derivative of Natural Logarithm (In)

To find the partial derivative of natural logarithm “In”, we have to proceed with the same procedure as finding the derivative of the normal function. But, here when we calculate the partial derivative of the function with respect to one independent variable taking other as constant and follow the same thing with others.

Also, see:

Partial Derivative Examples

Example 1: Determine the partial derivative of the function: f (x,y) = 3x + 4y.

Solution:

Given function: f (x,y) = 3x + 4y

To find ∂f/∂x, keep y as constant and differentiate the function:

Therefore, ∂f/∂x = 3

Similarly, to find  ∂f/∂y, keep x as constant and differentiate the function:

Therefore, ∂f/∂y = 4

Example 2: Find the partial derivative of f(x,y) = x2y + sin x + cos y.

Solution:

Now, find out fx first keeping y as constant

fx = ∂f/∂x = (2x) y + cos x + 0

= 2xy + cos x

When we keep y as constant cos y becomes a constant so its derivative becomes zero.

Similarly, finding fy

fy = ∂f/∂y = x2 + 0 + (-sin y)

= x2 – sin y

Example 3: Find  ∂f/∂x,  ∂f/∂y,  ∂f/∂z for the given function, f(x, y, z) = x cos z + x2y3ez

Solution:

To find  ∂f/∂x,  ∂f/∂y,  ∂f/∂z

Given Function:  f(x, y, z) = x cos z + x2y3ez

∂f/∂x = cos z + 2xy3ez

∂f/∂y = 3x2y2ez

∂f/∂z =-x sin z + x2y3ez

To learn more problems on partial dervatives, and the problems related to the differential equations, register with BYJU’S – The Learning App and download the app to learn all the important Maths-related concepts with ease.

 

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