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Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin (x + a) - Mathematics

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प्रश्न

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

sin (x + a)

योग
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उत्तर

Let f(x) = sin (x + a)

f(x + h) = sin (x + h + a)

By first principle,

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

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

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

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

= `lim_(h->0)1/h [( cos  (2x + 2a + h)/2)  {sin (h/2)/(h/2)}]`

= `lim_(h->0)1/h [((2x + 2a + h)/2)  lim_(h->0){sin (h/2)/((h/2))}]`     `["As"  h ->0 => h/2 ->0]`

= `cos  ((2x + 2a)/ 2) xx 1`       `[lim_(x->0) (sin x)/x = 1]`

= cos (x + a)

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अध्याय 12: Limits and Derivatives - Miscellaneous Exercise [पृष्ठ २५३]

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एनसीईआरटी Mathematics [English] Class 11
अध्याय 12 Limits and Derivatives
Miscellaneous Exercise | Q 14. | पृष्ठ २५३

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