In this article, we will again look at multivariable functions. We will consider how we can find tangent planes for functions of two variables, and how we can find tangent hyperplanes for functions of three or more variables. If you do not know the concepts of partial/directional derivates or the gradient, then I advise reading the following article before: Finding Derivatives of Multivariable Function. Now, let us get started.
Remembering Tangent Lines
Let us first remember what is meant by finding the tangent line for functions of a single variable, f(x). It can be defined in very simple words:
In simple words, the tangent line is the straight line that goes in the same direction as the curve in the given point, x = c. We can try to see it visually:
Defining Tangents for Multivariable Functions
We can then move on to finding the tangent when we consider multiple variables, whether it be with 2 or more variables. The tangent plane is then simply the plane through a point of a surface that contains the tangent lines to all the curves on the surface through the same point. In other words, it is the (hyper)plane created by considering the tangent lines in all the different directions in the given point.
