Let be a vector-valued function.
The derivative of is defined by:
…for any values of for which the limit exists. When the limit as exists, we say that is differentiable at .
The division in this definition is symbolic. As we can’t divide vectors, It means the scalar multiplication of by .
If we develop the limit given in this definition, we’ll find that, if , then:
Graphically, the derivative of a vector-valued function is the tangent vector to the function’s curve at .