Editing Derivative (section)

=== Vector-valued functions ===
A [[vector-valued function]] <math> \mathbf{y} </math> of a real variable sends real numbers to vectors in some [[vector space]] <math> \R^n </math>. A vector-valued function can be split up into its coordinate functions <math> y_1(t), y_2(t), \dots, y_n(t) </math>, meaning that <math> \mathbf{y} = (y_1(t), y_2(t), \dots, y_n(t))</math>. This includes, for example, [[parametric curve]]s in <math> \R^2 </math> or <math> \R^3 </math>. The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of <math> \mathbf{y}(t) </math> is defined to be the [[Vector (geometric)|vector]], called the [[Differential geometry of curves|tangent vector]], whose coordinates are the derivatives of the coordinate functions. That is,{{sfn|Stewart|2002|p=893}}
<math display="block"> \mathbf{y}'(t)=\lim_{h\to 0}\frac{\mathbf{y}(t+h) - \mathbf{y}(t)}{h}, </math>
if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of <math> \mathbf{y} </math> exists for every value of {{tmath|1= t }}, then <math> \mathbf{y}' </math> is another vector-valued function.{{sfn|Stewart|2002|p=893}}