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In this article, we will take a closer look at derivatives of multivariable functions. We will look at the Directional Derivative, the Partial Derivative, the Gradient, and the concept of C1-functions. In a future article, we will consider the concept of Hesse Matrices.
The Directional Derivative
Let us first consider the concept of directional derivatives. We can first see the formal definition, and afterward, we can try to understand it more intuitively. The formal definition is given by:
So, what does this say? Let us first remember that our function is now in multiple variables and not just a single one. For the ease of visualization, we assume that it is two variables. We have then gone from functions where we have two dimensions: x and f(x), to functions where we have three dimensions: x, y, and f(x,y). Let us take an illustrative example.
Imagine that we have a single variable function, f(x), which is defined as:
We then consider a two-variable function, f(x,y), which is defined as:
We have then added another dimension, which we can see with the visualization of our two graphs:
So, what does it then mean to take the directional derivative? Imagine that we have another multivariable function, which is defined as:
We then consider the point and the vector:
When we take the directional derivative, we then calculate what the rate of change is in the point, (1,0), going in the direction of the vector, (2,1). Remember that we have more dimensions now, so when we consider the rate of change in a point we now have multiple! We, therefore, need to consider in what direction we want to calculate the rate of change. Let us try to calculate this example. We remember that we have the following formula to use:
So, we first find:
We can then start the calculations:
We can then find the limit when h goes towards 0:
