Question

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Answer 1

We know that h is a vector-valued function. This means it takes one number as an input (t) ,but it outputs two numbers as a two-dimensional vector.

Finding the derivative of a vector-valued function is pretty straightforward. Suppose a vector-valued function is defined as
`u(t)=(v(t),w(t))` then its derivative is the vector-valued function
`u'(t)=(v'(t),w'(t))` .

In other words, the derivative is found by differentiating each of the expressions in the function's output vector.

Recall that
`h(t)=(2t^3+6,e^(2t))`

Let's differentiate the first expression:


`(d)/(dt) (2t^3+6)=3*2t^2) = 6t^2`

Let's differentiate the second expression:


`(d)/(dt)(e^2^t)= 2e^2^t`

Now let's put everything together:


`h′(t) = (d)/(dt)(2t^3+6),(d)/(dt)(e^2^t))`


`= 6t^2,2e^2^t`

In conclusion,
`h'(t) = (6t^2,2e^2^t)`