Find the derivative of a
vector-valued function 𝐫 of 𝑡 equals one plus 𝑡 cubed 𝐢 plus five 𝑡 squared
plus one 𝐣 plus 𝑡 cubed plus two 𝐤.
Remember, we can find the
derivative of a vector-valued function by taking the derivative of each
component. That means we’re individually going
to differentiate, with respect to 𝑡, one plus 𝑡 cubed, five 𝑡 squared plus one,
and 𝑡 cubed plus two. We then recall that to
differentiate polynomial terms, we multiply the entire term by the exponent and then
reduce that exponent by one. The derivative of one is zero, and
the derivative of 𝑡 cubed is three 𝑡 squared. So, differentiating our component
for 𝐢 and we get three 𝑡 squared.
Next, we’re going to differentiate
the component for 𝐣. That’s 10𝑡 plus zero, which is
simply 10𝑡. Finally, we’re going to
differentiate 𝑡 cubed plus two with respect to 𝑡. Of course, the derivative of that
constant is zero, so we obtain the derivative of 𝑡 cubed plus two to be three 𝑡
squared plus zero or just three 𝑡 squared. So, the derivative 𝑟 prime of 𝑡
is three 𝑡 squared 𝐢 plus 10𝑡 𝐣 plus three 𝑡 squared 𝐤.