Question

Find the derivative of the following function:

cosec x

Answer 1

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