1. Why is it important to know about differentiating from first principles?
Knowing how to differentiate from first principles indicates some level of understanding of the principles behind the process rather than just knowing a quick rule.
When solving problems involving calculus you would use these quicker rules for differentiating and these rules will be learned in subsequent topics.
2. Can trigonometric functions be differentiated?
Yes, all functions are differentiable except for some at a few points where they fail to meet certain criteria.