Suppose we have a function of two variables.
We have the variable X.
We have the variable Y.
And we have Z, which is a function of both X and Y.
Suppose that we pick one value for X,
and we keep X at this one value as we change the value for Y.
Each time we change Y by a small amount, Z also changes by a small amount.
At each point,
the change in Z divided by the change in Y is given by the slope of this line.
Red indicates a negative slope.
Let’s now pick a different value for X, and keep X at this new value.
Again, at each point,
the change in Z divided by the change Y is given by the slope of this line.
The change in Z divided by the change in Y is what we
refer to as the partial derivative of Z with respect to Y.
Every point on the graph has a value for
the partial derivative of Z with respect to Y.
Here, green indicates a positive value, and red indicates a negative value.
Every point on the graph also has a value for
the partial derivative of Z with respect to X.
In this case, the value for Y is held constant, while the value for X changes.
Suppose we represent the slope of this line with an arrow.
The length and direction of the arrow indicates
if the slope is positive or negative, and by how much.
Every point has an arrow that represents
the partial derivative of Z with respect to X.
Every point also has an arrow that represents
the partial derivative of Z with respect to Y.
If we add these two arrows together as vectors, then we will get a new arrow.
Let us call this new arrow the gradient of the function Z.
This arrow always indicates
the direction in which Z is increasing by the largest amount.
The length of this arrow indicate by how much Z is increasing in this direction.
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