Question

Prove using the first principal that `d/dx (sqrtcosx) = -sinx/ (2(sqrtcosx)` ?

Answer 1

Please see below.

Explanation:

As `f(x)=sqrt(cosx)`, `f(x+h)=sqrt(cos(x+h))` and

`(dy)/(dx)=lim_(h->0)(f(x+h)-f(x))/h`

= `lim_(h->0)(sqrt(cos(x+h))-sqrtcosx)/h`

= `lim_(h->0)(sqrt(cos(x+h))-sqrtcosx)/hxx(sqrt(cos(x+h))+sqrtcosx)/(sqrt(cos(x+h))+sqrtcosx)`

= `lim_(h->0)(cos(x+h)-cosx)/(h(sqrt(cos(x+h))+sqrtcosx))`

= `lim_(h->0)(-2sin(x+h/2)sin(h/2))/(h(sqrt(cos(x+h))+sqrtcosx))`

= `lim_(h->0)(sin(h/2)/(h/2))xxlim_(h->0)(-sin(x+h/2))/(sqrt(cos(x+h))+sqrtcosx)`

= `1xx(-sinx)/(2sqrt(cosx))`

= `-sinx/(2sqrt(cosx))`