Earlier, we saw that functions are differentiated by using the limit of the variable in the quotient. But vectors, as we know, are not like scalars in that we cannot divide by vectors, which creates the need for new definitions for vector-valued functions.
We can define a vector function as a function 
—that is, it takes in a scalar value as input and outputs a vector. So, the derivative of F is defined as follows:
In the preceding equation, δx is a small perturbation on x. Additionally, F is only differentiable if the following applies: ...
