Answer :
Let f(x) = 3 cot x + 5 cosec x
f’(x) = 3 (cot x)’ + 5 (cosec x)’
Let f1(x) = cot x, accordingly f1(x + h) = cot (x + h)
By first principle
=
= 
=
= 
= 
∴ (cot x)’ = - cosec2 x ………………..(2)
Let f2(x) = cosec x, accordingly f2(x + h) = cosec (x + h)
By first principle
=
=
= 
= 
= 
= -cosec x cot x
∴ (cosec x)’ = -cosec x cot x
So, f’(x) = 3 (cot x)’ + 5 (cosec x)’
Putting (cot x)’ and (cosec x)’ in f’(x)
f’(x) = 3 × (-cosec2 x) + 5 × (-cosec x cot x)
f’(x) = -3cosec2 x – 5cosec x cot x
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