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Section 12.7 : Calculus with Vector Functions

6. Compute the derivative of the following limit.

\[\vec r\left( t \right) = \left\langle {\frac{{t + 1}}{{t - 1}},\tan \left( {4t} \right),{{\sin }^2}\left( t \right)} \right\rangle \] Show Solution

There really isn’t a lot to do here with this problem. All we need to do is take the derivative of all the components of the vector.

\[\begin{align*}\vec r'\left( t \right) & = \left\langle {\frac{{\left( 1 \right)\left( {t - 1} \right) - \left( {t + 1} \right)\left( 1 \right)}}{{{{\left( {t - 1} \right)}^2}}},4se{c^2}\left( {4t} \right),2\sin \left( t \right)\cos \left( t \right)} \right\rangle \\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{\left\langle {\frac{{ - 2}}{{{{\left( {t - 1} \right)}^2}}},4se{c^2}\left( {4t} \right),2\sin \left( t \right)\cos \left( t \right)} \right\rangle }}\end{align*}\]

Make sure you haven’t forgotten your basic differentiation formulas such as the quotient rule (the first term) and the chain rule (the third term).