Partial Differentiation:
In the terms of calculus, if we have a function with several variables like x,y then the derivative of that function with respect to some certain variable i.e. “x” while the “y” will remain constant is called the partial derivative of that function.
Standard Derivative Definition:
Partial differentiation is what happens when you differentiate a function of more than one variable. Let's start by considering the Standard derivative,
So if i want to know dy/dx for a function of f(x), then this may be
$$ \frac{df}{dx} = \lim\limits_{h \to 0} \left( \frac{f(x+h)-f(x)}{h} \right) $$
If we were to look at the graph of this function which would be something like the following:
So, we're working out how our function f changes in the “x” direction,because f is just a line.
What this means is that f can only Change in the x direction so as we move along the x-axis we're interested in how the value of “f” or the value on the y-axis changes.
Introduction of Partial Differentiation:
So, the only real derivative we can do here is the f/dx because that is the only direction in which “f” is changing.
Now if we have a function of several variables and let's just start with two variables e.g
we have f(x,y).
So, Now “f” depends on both “x and y” and so what this is actually going to give you is a surface.
So, if you were to plot z=f(x,y) you get a three dimensional surface which will change in both the x direction and the y direction. so if we consider the partial derivative example
$$ z=xy^2+yx^3 $$
Now we plot it, it will be a 3D or three dimensional graph. Let’s see a computer generated graph which we obtained here
we can see that it is indeed a surface it's changing in both the x and the y directions simultaneously. So, therefore it now makes sense that you might want to know how the function is changing in more than one direction as an example from this particular example.
Formal Definition of Partial Derivative:
What partial derivatives really do is allow us to calculate the rate of Change for a given direction.
So the x partial differentiation tells us how our function f(x,y) changes specifically moving in the x direction and then the y partial derivative will tell us how our function changes if we only move in the y direction.
So mathematically our definitions are going to be as follows:
If I want to know the partial x derivative of f where f is a function of x and y.
$$ \frac{∂f}{∂x} = \lim\limits_{h \to 0} \left( \frac{f(x+h,y)-f(x,y)}{h} \right) $$
So you'll notice it's very similar in fact almost identical to our full or standard derivative.
But in the partial case what we're doing is we're adding our h our small increment this is added only to the x coordinate of our function y remains unchanged.
So, this is telling us that the y-coordinate doesn't change the value of y remains constant and we're just looking at what happens if we change the x value only.
The partial y derivative is of course very similar so here we're interested in how f Changes in the y direction. So, it will be:
$$ \frac{∂f}{∂x} = \lim\limits_{h \to 0} \left( \frac{f(x,y+h)-f(x,y)}{h} \right) $$
So, very similar to the partial x except now we do the plus h on the y coordinate because we're interested in how the function changes when we change y. So, the x coordinate here remains constant.
Partial Derivative Notation:
Here, you see first of all we use these curly "∂"" notation which denotes the standard form of writing partial derivative.
The partial derivative notation is given as,
$$ \frac{∂f}{∂x} $$
Partial Derivative Examples:
Let's actually work through an example of the function we looked at earlier :
$$ f(x,y)=xy^2+yx^3 $$
So, Let's compute the partial derivative example to see how this works in practice. As there are two variables "x" and "y" in above example, So we have to find the partial derivative with respect to x and y. The key thing here is to remember that :
When you're doing your x partial derivative y doesn't change so that means you can basically treat y as a constant.
Similarly, when you do your y partial derivative x isn't changing so therefore you treat x as a constant.
Let’s Calculate partial derivative with respect to x:
So our usual rules of differentiation apply: we still differentiate any function of x as though it is a full or standard derivative, however remember y is constant in the x partial.
$$ \frac{∂f}{∂x} = y^2+3yx^2 $$
Let’s Calculate partial derivative with resppect to y:
y partial will be done using the same method we only differentiate y terms by standard method of derivatives and treating x as constant like some numbers.
$$ \frac{∂f}{∂y} = 2xy+x^3 $$
So, this is a basic concept of partial derivatives. You can also go further to the second order partial derivative here which is also an important concept. so, practice by yourself and make it interesting for yourself.