Derivative
Vector-valued functions

A vector-valued function y {\displaystyle \mathbf {y} } of a real variable sends real numbers to vectors in some vector space R n {\displaystyle \mathbb {R} ^{n}}. A vector-valued function can be split up into its coordinate functions y 1 (t), y 2 (t), …, y n (t) {\displaystyle y_{1}(t),y_{2}(t),\dots,y_{n}(t)}, meaning that y = (y 1 (t), y 2 (t), …, y n (t)) {\displaystyle \mathbf {y} =(y_{1}(t),y_{2}(t),\dots,y_{n}(t))}. This includes, for example, parametric curves in R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}}. The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of y (t) {\displaystyle \mathbf {y} (t)} is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is, y ′ (t) = lim h → 0 y (t + h) − y (t) h, {\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},} if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of y {\displaystyle \mathbf {y} } exists for every value of t {\displaystyle t}, then y ′ {\displaystyle \mathbf {y} '} is another vector-valued function.
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