Question

Find the derivative of the vector function

Short answer

The derivative of the vector function is

\[r'(t) = \left( {{{\sec }^2}t,\sec t\tan t, - \frac{2}{{{t^3}}}} \right)\]

Step-by-step solution

Step 1: Applied the derivative function

Use the formula for derivative

\[r'(t) = \frac{d}{{dt}}(r(t))\]

Step 2: Solution of the vector function

Let’s applied the formula

\[\begin{array}{l}r'(t) = \frac{d}{{dt}}\left( {\tan t,\sec t,\frac{1}{{{t^2}}}} \right)\\\tan t = {\sec ^2}t\\\sec t = \sec t\tan t\\\frac{1}{{{t^2}}} = - \frac{2}{{{t^3}}}\\r'(t) = \left( {{{\sec }^2}t,\sec t\tan t, - \frac{2}{{{t^3}}}} \right)\end{array}\]