Geometric Interpretation of Partial Derivatives

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We’ve defined the partial derivatives of a function as follows.

Consider a function $f:\mathbb {R}^n\rightarrow \mathbb {R}$ . For $1\leq i\leq n$ , we define the partial derivative of $f$ with respect to $x_i$ to be $f_{x_i}(x_1,...,x_n) = \lim _{h\rightarrow 0}\frac {f(x_1,...,x_{i-1}, x_i+h, x_{i+1},...,x_n)-f(x_1,...,x_n)}{h},$ provided this limit exists.

In other words, if we treat all variables except for $x_i$ as constants, and differentiate with respect to $x_i$ , we get the partial derivative with respect to $x_i$ .

When computing a partial derivative with respect to $x_i$ , we’re looking at the instantaneous rate of change of $f$ with respect to $x_i$ , if we keep the rest of the variables constant. Roughly speaking, we’re asking: how does increasing $x_i$ a tiny bit affect the value of $f$ ?

We can see the partial derivatives reflected in the shape of the graph of $f$ . So that we can visualize the graph of $f$ , we’ll focus on a function $f:\mathbb {R}^2\rightarrow \mathbb {R}$ , so we’re considering the partial derivative of $f$ with respect to $x$ , and with respect to $y$ .

Suppose at the point $(1,2)$ , we have that $f_x(1,2)>0$ and $f_y(1,2)>0$ . Then, around the point $(1,2)$ , if we move a tiny amount in the positive $x$ direction, the value of $f$ will increase. If we move a tiny amount in the positive $y$ direction, the value of $f$ will increase as well.

Similarly, suppose at the point $(1,2)$ , we have that $f_x(1,2)<0$ and $f_y(1,2)<0$ . Then, around the point $(1,2)$ , if we move a tiny amount in the positive $x$ direction, the value of $f$ will decrease. If we move a tiny amount in the positive $y$ direction, the value of $f$ will decrease as well.

Now, let’s consider the case where $f_x(1,2)>0$ and $f_y(1,2)<0$ . Then, around the point $(1,2)$ , if we move a tiny amount in the positive $x$ direction, the value of $f$ will increase. But, if we move a tiny amount in the positive $y$ direction, the value of $f$ will decrease.

Next, let’s suppose that $f_x(1,2)>0$ and $f_y(1,2) =0$ . As expected, $f$ increases as we move a tiny amount in the positive $x$ direction. On the other hand, the graph of $f$ has flattened out as we move in the $y$ direction. However, this doesn’t mean that it’s constant! It’s just the instantaneous rate of change that’s $0$ at that one point.

Now, let’s look at a case where $f_x(1,2)=0$ and $f_y(1,2)=0$ . As before, this does not mean that $f$ is constant. This just means that the rates of change are both instantaneously $0$ . Points with this property will be important later in the course, when we study optimization.