define f'(x)

Question

f(x)=cosx-sinx.cosx+sin²x.cosx-sin³x.cosx+...

what is f'(x) if 0<x<π/2 ?

Answer 1

f(x)=cosx-sinx.cosx+sin²x.cosx-sin³x.cosx+...
what is f'(x) if 0<x<π/2 ?

 

f(x) can be factored: RHS is cos x times [1 - sin x + sin²x - sin³x +...]

 

and I recognize the bracketed expression pattern as

   1/(1+t)=1−t+^t2−t³+? 

   if |t|<1 (infinite geometric series).

 

So f(x) = (cos x) / ((1+sin x).  Now you can differentiate this according to the Quotient Rule.