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प्रश्न
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
cos x at x = `(5pi)/4`
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उत्तर
Let f(x) = cos x
∴ `"f"((5pi)/4) = cos ((5pi)/4)`
`"f"((5pi)/4 + "h") = cos ((5pi)/4 + "h")`
∴ `"f"((5pi)/4 + "h") - "f"((5pi)/4)`
= `cos((5pi)/4 + "h") - cos ((5pi)/4)`
= `-2sin [((5pi)/4 + "h" + (5pi)/4)/2] sin [((5pi)/4 + "h" - (5pi)/4)/2]`
= `-2 sin [((5pi)/2 + "h")/2] sin ("h"/2)`
By definition,
`"f'"((5pi)/4) = lim_("h" -> 0) ("f"((5pi)/4 + "h") - "f"((5pi)/4))/"h"`
= `lim_("h" -> 0) (-2sin [((5pi)/2 + "h")/2] sin("h"/2))/"h"`
= `lim_("h" -> 0) [-2 sin [((5pi)/2 + "h")/2] (sin("h"/2))/(("h"/2))] xx 1/2`
= `-[lim_("h" -> 0) sin [((5pi)/2 + "h")/2] ] xx [lim_("h" -> 0) (sin("h"/2))/("h"/2)]`
= `-sin [((5pi)/2 + 0)/2] xx 1 ...[because "h" -> 0, "h"/2 -> 0 "and" lim_(theta -> 0) sintheta/theta = 1]`
= `- sin ((5pi)/4)`
= `- sin (pi + pi/4)`
= `-(- sin pi/4)`
= `1/sqrt(2)`
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