Question

If \(f\left( t \right) = \sec t\), find \(f''\left( {\frac{\pi }{4}} \right)\).

Short answer

The required value is \(f''\left( {\frac{\pi }{4}} \right) = 3\sqrt 2 \).

Step-by-step solution

Differentiating given function

Thegiven function is:\(f\left( t \right) = \sec t\)

Differentiating the given function twice, with respect to\(t\), we get:

\(\begin{aligned}f'\left( t \right) &= \frac{d}{{dt}}\left( {\sec t} \right)\\ &= \sec t\tan t\\\\f''\left( t \right) &= \frac{d}{{dt}}\left( {\sec t\tan t} \right)\\ &= \sec t\frac{d}{{dt}}\left( {\tan t} \right) + \tan t\frac{d}{{dt}}\left( {\sec t} \right)\\ &= {\sec ^3}t + \sec t{\tan ^2}t\end{aligned}\)

On putting given value for t

Now, the value for \(f''\left( {\frac{\pi }{4}} \right)\) ,we have:

\(\begin{aligned}f''\left( t \right)\left| {_{t = \frac{\pi }{4}}} \right. & = {\sec ^3}\left( {\frac{\pi }{4}} \right) + \sec \left( {\frac{\pi }{4}} \right){\tan ^2}\left( {\frac{\pi }{4}} \right)\\ & = 2\sqrt 2 + \sqrt 2 \\ &= 3\sqrt 2 \end{aligned}\)

Hence, the required value is \(f''\left( {\frac{\pi }{4}} \right) = 3\sqrt 2 \).