Derivatives of single-variable functions, such as , are commonly represented by notations like or . For functions involving multiple variables, like or , it becomes essential to specify the exact variable for differentiation, keeping all other variables constant.
To distinguish partial derivatives from the ordinary derivatives we saw earlier, we use a different symbol: . This symbol is often called the "curly d," "del," or simply "partial." It signals that we are performing a partial differentiation.
The most common notation you'll encounter, similar to the notation for ordinary derivatives, uses the symbol.
If we have a function , its partial derivative with respect to is written as:
This notation reads as "the partial derivative of with respect to ." It explicitly tells us:
Similarly, the partial derivative of with respect to is written as:
If the function output is assigned to another variable, say , you might also see the notation written as or .
Another common and often more compact notation uses subscripts to indicate the variable of differentiation. For the same function :
This notation is especially convenient when writing out the gradient vector, which we'll see shortly.
Let's consider a simple function .
(We'll learn how to calculate these in the next section, but for now, focus on understanding what the notation represents).
Just like with ordinary derivatives, we often want to evaluate a partial derivative at a specific point. If we want to find the partial derivative of with respect to at the point , we can write it as:
All these notations mean the same thing: calculate the partial derivative expression with respect to , and then substitute and into the result.
Understanding this notation is fundamental because it's the language used to describe how complex functions, like the cost functions in machine learning models, change as their individual inputs or parameters are adjusted.
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