Question

1) Find d/dt ( sec t/2)

Answer 1

`(d)/(dt) [sec((t)/(2) )] = (1)/(2) sec((t)/(2) )tan((t)/(2) )`

General Formulas and Concepts:

Calculus

• Chain Rule:
  `(d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`
• Trig u derivative:
  `(d)/(dt) [sec(u)] = u'[sec(u)tan(u)]`

Explanation:

Step 1: Define

`(d)/(dt) [sec((t)/(2) )]`

Step 2: Differentiate

1. Trig u [Chain Rule/Basic Power]:
   `(d)/(dt) [sec((t)/(2) )] = (t^(1-1))/(2) sec((t)/(2) )tan((t)/(2) )`
2. Simplify:
   `(d)/(dt) [sec((t)/(2) )] = (t^(0))/(2) sec((t)/(2) )tan((t)/(2) )`
3. Evaluate:
   `(d)/(dt) [sec((t)/(2) )] = (1)/(2) sec((t)/(2) )tan((t)/(2) )`