What is the significance of partial derivative? Give an example and help me to understand in brief.
I hope it helps.
The partial derivative is intrinsically associated to the total variation.
Suppose we have a function
Fixing ideas, making
In our function-example we have
so we can compute the total variation for a given function, by calculating the partial derivatives
Here, the quantities
In our example
To supplement Cesareo's answer above, I will provide a less mathematically rigorous introductory definition.
The partial derivative, loosely speaking, tells us how much a multi-variable function will change when holding other variables constant. For instance, suppose we are given
Suppose the company which manufactures the product would like to know how much more utility they can get out of it if they increase the lifespan of the product by 1 unit. The partial derivative will tell the company this value.
The partial derivative is generally denoted by the lowercase Greek letter delta (
If we're trying to find how much the utility of the product changes with a 1 unit increase in time, we are computing the partial derivative of utility with respect to time:
To compute the PD, we hold other variables constant. In this case, we treat
Thus, a 1 unit increase in the time the product is used produces
There is much, much more to be said about partial derivatives - in fact, entire undergraduate and graduate courses can be devoted to solving just a few types of equations involving partial derivatives - but the basic idea is that the partial derivative tells us how much one variable changes when the other ones remain the same.