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प्रश्न
Find the derivative of the following function:
cosec x
उत्तर
Let f(x) = cosec x
and f(x + h) = cosec (x + h)
f(x + h) – f(x) = cosec (x + h) – cosec x
= `1/(sin (x + h)) - 1/(sin x)`
= `(sin x - sin (x + h))/(sin (x + h) sin x)`
= `(-2 cos(x + h/2) sin h/2)/(sin (x + h) sin x)`
`f'(x) = lim_(h → 0)(f(x + h) - f(x))/h`
= `lim_(h → 0)1/h. (-2cos (x + h/2) sin h/2)/(sin (x + h) sin x)`
= `lim_(h → 0)(- cos(x + h/2))/(sin (x + h)sin x). ((sin h/2)/(h/2))`
= `-cosx/(sin x. sin x)`
= `- 1/sin x. (cos x)/(sin x)`
= − cosec x cot x
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