Gradient for 2-dimension (x-y)
In simpler terms, a gradient
is a slope. You can calculate a function’s first-order derivative to get its gradient. This tutorial explains gradients with an example and plots. For instance, the following is a function of x, and the function can be plotted on a two-dimension x-y
space.
\[ \ y=x^2+6x+10 \]

If you calculate its first-order derivative, you can get the following function.
\[ \ \frac{dy}{dx}=2x+6 \]
- For
x = -3
, its slope is 0. That is, the gradient is 0 for the point of(-3, 1)
.
- For
x = 5
, its slope is 16. That is, the gradient is 16 for the point(5, 65)
.
What does the slope of 16 mean? It means for the change of 1 unit on the x-axis, the y-axis is 16. Literally, it is the definition of slope.

\[ \ \frac{\triangle y}{\triangle x}=16 \]
Gradient for 3-dimension (x-y-z)
For a function with 2 independent variables (x, y)
and 1 dependent variable (z)
, it will be 3-dimension if we plot it (x-y-z
). In this case, the gradient still is the slope, but such a slope is determined by 2 parameters or factors (i.e., x and y). The following is an example of 3-dimension function.
\[ \ z=f(x, y) = x^2+y^2 \]
If we plot it, it looks as follows.

Its gradient is as follows.
\[ \nabla f(x, y)=\left[ \begin{array} {} 2x \\ 2y \end{array} \right] \]
Since we get the function of the gradient, we can get a specific gradient for any point. For instance, for the point of (4, -6), its gradient is as follows.
\[ \nabla f(2, 3)=\left[ \begin{array} {} 8 \\ -12 \end{array} \right] \]
However, in such a 3-dimensional space, it is difficult to visually understand what the slope would look like. This is a bit different from the 2-dimensional space shown earlier.
Gradient for more than 3-dimension
From a more generalized perspective (i.e., with more than 2 IVs), we can write its gradient as a vector, a combination of all the partial derivatives. To calculate the gradient, you just need to insert the value of p(x, y, w...)
to the vector.
\[ \nabla f(p)=\left[ \begin{array} {} \frac{\partial f}{\partial x} f(p) \\ \frac{\partial f}{\partial y} f(p) \\\frac{\partial f}{\partial w} f(p) \\ … \end{array} \right] \]
The following is an example.
\[ \ z=f(x, y, w) = x^2+y^2+w^2 \]
Its gradient is as follows.
\[ \nabla f(p)=\left[ \begin{array} {} 2x \\ 2y \\ 2w \end{array} \right] \]
However, it is very difficult to plot more than 3 dimensions, if not impossible.
Further Reading
There are a few posts on the Internet about this topic, but the one I found useful is written by Robert Kwiatkowski. My write-up is inspired by his writing, with some expanded details. I hope you find this tutorial useful. I also have another tutorial about this topic, see below.