What does it mean to differentiate a vector?I didn’t quite understand what does it mean to differentiate a vector, i suspect that the derivative of a vector valued function is just answering the question : What vector should i added to the previous one to get the next one. For example :Consider the vector X → ( t ) = ⟨ 1 , t ⟩ X → ′ ( t ) = ⟨ 0 , 1 ⟩Now if i take a random vector from the vector valued function ⟨1,t⟩ , say for example ⟨1,3⟩ If i add to it the derivative of ⟨1,t⟩ which is ⟨0,1⟩, i get ⟨1,4⟩ which is the next vector.But this is only true for t ∈ Z because in the real numbers the next vector is not defined, it could be ⟨ 1 , 3.0001 ⟩So how can i really understand what is the derivative of a vector.

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What does it mean to differentiate a vector?I didn’t quite understand what does it mean to differentiate a vector, i suspect that the derivative of a vector valued function is just answering the question : What vector should i added to the previous one to get the next one. For example :Consider the vector            X      →        (  t  )  =  ⟨  1  ,  t  ⟩                    X        →              ′    (  t  )  =  ⟨  0  ,  1  ⟩Now if i take a random vector from the vector valued function ⟨1,t⟩ , say for example ⟨1,3⟩ If i add to it the derivative of ⟨1,t⟩ which is ⟨0,1⟩, i get ⟨1,4⟩ which is the next vector.But this is only true for   t  ∈      Z    because in the real numbers the next vector is not defined, it could be   ⟨  1  ,  3.0001  ⟩So how can i really understand what is the derivative of a vector.

Hamnetmj · Accepted Answer

One way to think of it is by components. A vector in n dimensions has n components, and each of those can be a function of one or more variables. So if you have a vector       v    →    =&amp;lt;  x  (  t  )  ,  y  (  t  )  ,  z  (  t  )  &amp;gt;, then its derivative would be             v      →        ′    =&amp;lt;      x    ′    (  t  )  ,      y    ′    (  t  )  ,      z    ′    (  t  )  &amp;gt;Vectors don&#039;t have to have integer components, and neither do their derivatives. So you don&#039;t have to worry about the &quot;next&quot; vector any more than you have to worry about the &quot;next&quot; number when dealing with   y  =  f  (  x  )

beefypy · Accepted Answer

X          ′                (        t        )                            =                  lim                      h            →            0                                                X            (            t            +            h            )            −            X            (            t            )                    h                                              lim                      h            →            0                          X        (        t        +        h        )                            =        X        (        t        )        +                  lim                      h            →            0                                    X          ′                (        t        )        ⋅        h            or in other words, as   h  →  0,   X  (  t  +  h  )  =  X  (  t  )  +      X    ′    (  t  )  ⋅  hIn your calculation of &quot;next vector&quot;, I think you missed multiplying by h?

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