Overview
We use partial differentiation in vector calculus and differential geometry.
When the function depends on two or more variables then the derivative changes into partial derivative.
A partial derivative of a function of several variables is an ordinary derivative with respect to one of the variables when all the other variables remain constant.
We use all the rules of differentiation but the condition is that if we are differentiating partially with respect to one variable, all the other variables are treated as constants.
Definition of Partial Differentiation
Suppose we have a function f(x, y), which is dependent on two variables x and y. x and y are independent of each other.
Then we will say that the function ‘f’ partially depends on these two variables.
The derivative of the function ‘f’ is called the partial derivative of ‘f’.
Notations
Suppose we have a function u = f (x, y) of two independent variables x and y. If we keep y constant and x varies then function u becomes a function of x only. The derivative of u with respect to x, keeping y as constant is called partial derivative of ‘u’, with respect to ‘x’.
The following notations we use for partial differentiation-
Similarly, the partial derivative of ‘u’ with respect to ‘y’ keeping x as constant is denoted by
Product and quotient rules of partial derivative
Product rule:
If u = f(x, y).g(x, y), then
And
Quotient rule:
If u = f(x, y)/g(x, y), then [note- g(x, y) should be non-zero]
And
Partial derivative of higher orders
Let u = f(x, y) then 
These are called the second-order partial derivatives.
Note-
Solved examples
Example: Differentiate the following function partially with respect to x and y.
Solution:
Here the function is-
Now differentiating partially with respect to ‘x’, we get-
Similarly differentiating partially with respect to ‘y’, we get-
Example: Find the value of of the following function-
Solution:
Here we have-
First we will partially differentiate the given function with respect to ‘z’
Now
Finally
Which can be written as-
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