Partial Derivative

What is Partial Differentiation?

Partial derivatives are used in vector analysis and differential geometry. While in vector analysis and geometry, there are different functions given in the question, some of the functions depend on two or more variables. By contrast, in some cases, there's just one variable. 

Hence, If a function depends on only one variable, then its derivative is known as 'ordinary differentiation.' At the same time, if a function depends on more than one variable, then its derivative, taking with respect to either of the two variables, is known as 'partial differentiation.' Also, the derivative which covers all the variables is known to be 'total differentiation.' 

What is The Definition of Partial Differentiation?

Suppose there's a function, and its variables are x and y; it will be represented as f(x, y), so, over here, f is dependent on x and y, whereas both the variables are independent of each other.

For example, if a person is going to buy a new house, the variable will be the cost of labour, the cost of materials, and the cost of land. As in this situation, there are three variables, if the value of one variable increases; for example, the cost of land, then the price of the house will also increase. 

Thus, the house is dependent on the other three variables, and we can say that the function is partially dependent, but the cost of labour, material, and land are independent of each other.

Furthermore, if we differentiate the function f with respect to x, then y will be constant, and if y is taken as a variable, then x will be constant. 

Symbolic Representation of Partial Differentiation 

The partial derivative symbol is a swirly 'd,' ∂ and it's called dell. The primary reason behind representing the partial derivative with a swirly d, is because all the other derivatives are represented by, d, and therefore one can differentiate partial derivatives easily. 

Example:  Suppose the question is f(x, y), where f is the function of x and y. Hence, the partial derivative of f concerning x will be represented as ∂f/∂x, and the other variable y will be kept constant while one should clearly remember that ∂x,  dx are not the same. 

Definition for Partial Derivative

Suppose, if f(x, y) is the function, wherein f partially depends on both x and y, and hence if differentiated f with respect to x and y, then the derivative will be called the partial derivative of f.  The partial derivative formula of, f with respect to both the variable x and y will be given as:

fx   =  \[\frac{{df}}{dx}\]  =   \[\lim_{h_{\rightarrow }0}\]  \[\frac{{f(x + hy) - (f(xy)}}{h}\] 

And partial derivative of function f with respect to y, keeping x as constant, we get;

 fy =  \[\frac{{df}}{dx}\]  =   \[\lim_{h_{\rightarrow }0}\]  \[\frac{{f(xy + h) - (f(xy)}}{h}\] 

Partial Derivative Rules

Just like the ordinary derivative, there is also a different set of rules for partial derivatives. Rules for partial derivatives are product rule, quotient rule, power rule, and chain rule.

Product Rule for the Partial Derivative

If u = f(x,y).g(x,y), then the product rule states that:

u_x   = \[\frac{{du}}{dx}\] = g(x,y) \[\frac{{df}}{dx}\] + f(x,y) \[\frac{{dg}}{dx}\]

And, uy   =   \[\frac{{du}}{dy}\] = g(x,y) \[\frac{{df}}{dy}\] + f(x,y) \[\frac{{dg}}{dy}\]

Quotient Rule for the partial derivative 

Suppose, u = f(x,y)/g(x,y), where g(x,y) ≠ 0, then the quotient rule states that:

 ux =   \[\frac{{g(xy)\frac{df}{dx} - f(xy)\frac{dg}{dx}}}{[g(xy)]^{2}}\]

And uy =   \[\frac{{g(xy)\frac{df}{dy} - f(xy)\frac{dg}{dy}}}{[g(xy)]^{2}}\]

Power Rule

Given that, u = [f(x, y)]n then, the partial derivative of u with respect to x and y will be formulated as:

 ux = n|f(x,y)|n-1  ∂f/∂x

And uy = n|f(x,y)|n-1   ∂f/∂y

Partial Derivative Chain Rule

The chain rule differs for one independent variable and two independent variables, and are given as:

Chain Rule for The Independent Variable

Suppose x=g(t) and y=h(t) are differentiable functions of t, and  z = f(x, y) is a differentiable function of both x and y. Then, z can be written as z = f(g(t), h(t)) - a differentiable function of t. The partial derivative of the function with respect to the variable t will be given as follows:

∂z/∂t = ∂z/∂x × ∂x/∂t + ∂z/∂y × ∂y/∂t

In the above-given situation, the ordinary derivatives are assumed at t, but the partial derivatives are evaluated at (x, y).

Chain Rule for The Dependent Variable

Assuming x = g (u, v) and y = h (u, v) are the differentiable functions of the given variables u and v, and similarly, z = f (x, y) is a differentiable function of x and y. Thus z can be further defined as z = f (g (u, v), h (u, v)), which is a differentiable function both of u and v. Therefore, the partial derivative of the above functions with respect to the given variables will be stated as:

∂z/∂u = ∂z/∂x × ∂x/∂u + ∂z/∂y × ∂y/∂u

And,

∂z/∂v = ∂z/∂x × ∂x/∂v + ∂z/∂y × ∂y/∂v

The Partial Derivative of Natural Logarithm (In)

For finding the partial derivative of natural logarithm "In," the procedure is the same as finding the derivative of any normal function. However, the partial derivative of the function is calculated with respect to one independent variable, and the others are taken as constant. 

Partial Derivative examples

Below given are some partial differentiation examples solutions:

Example 1. Determine the partial derivative of the function: f(x, y)=4x+5y

Solution:  The function provided here is f (x,y) = 4x + 5y

To find ∂f/∂x, we have to keep y as a constant variable, and differentiate the function:

Therefore, ∂f/∂x will be 4

Likewise, to find  ∂f/∂y, we have to keep x as a constant variable and differentiate the function:

Therefore, ∂f/∂y will be 5.

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

Solution: Now, to find out fx, we have to first keep the variable y as constant.

Hence, fx = ∂f/∂x = (2x) y + cos x + 0

= 2xy + cos x

When y is kept as a constant variable, cos y also becomes a constant. Thus, the derivative will turn out to be 0.

Similarly, for, fy

fy = ∂f/∂y = x2 + 0 + (-sin y)

= x2 – sin y.