Question

Find the derivative of the following function:

3cot x + 5cosec x

Answer 1

(x) = cot x

f(x + h) = cot (x + h)

f(x + h) – f(x) = cot (x + h) – cot x

= `(cos (x +h))/(sin (x + h)) - cos x/sin x`

= `(cos (x +h) sin x - cos x sin (x +h))/(sin (x +h) sin x)`

= `(-[sin (x +h) cosx - cos(x +h) sin x])/(sin (x +h) sin x)`

= `(sin (x + h - x))/(sin (x + h)sin x)`

= `- sin h/(sin (x +h) sin x)`

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

= ` lim_(h → 0) 1/h. [sin h/(sin (x +h)sin x)]`

= `-1. 1/(sin x sin x)`

= `- cosec^2 x`

Now, f'(x)= `3 d/dx cotx + 5 d/dx cosec x` 

= 3(- cosec² x) + 5 (- cosec x.cot x)

= – 3 cosec²x – 5 cosec x cot x