Question

Find the derivative of the following function from first principle: 

`cos (x - pi/8)`

Answer 1

Let f(x) = `cos (x - pi/8)` Accordingly, f(x + h) = `cos (x + h - pi/8)`

By first principle,

(f(x) = `lim_(h->0) (f(x + h) - f(x))/h`

= `lim_(h->0) (cos (x + h - pi/8) - cos (x - pi/8))/h`

= `lim_(h->0) [-2 sin  ((x + h - pi/8 + x - pi/8)/2) sin  ((x + h - pi/8 -x pi/8)/2)]`

= `lim_(h->0) 1/h [-2 sin ((2x + h - pi/4)/2) sin  h/2]`

= `lim_(h->0)[- sin ((2x + h - pi/4)/2) (sin  (h/2))/((h/2))]`

= `lim_(h->0) [- sin ((2x + h - pi/4)/2)]` `lim_(h->0) (sin  (h/2))/((h/2))`      `["As"  h -> 0 => h/2 -> 0]`

= `- sin ((2x + 0 - pi/4)/2).1`

= `-sin (x - pi/8)`