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प्रश्न
Find the derivative of the following function from first principle:
`cos (x - pi/8)`
उत्तर
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)`
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