Differentiate each of the following from first principle: `sinx+cos x`

Explanation

`f(x)=sinx+cos x`


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


`f'(x)=lim_(h->0)(sin(x+h)+cos (x+h)-sinx-cos x)/(h)`


`f'(x)=lim_(h->0)(sin(x+h)-sinx+cos (x+h)-cos x)/(h)`


`f'(x)=lim_(h->0)(2cos(x+h/2)sin(h/2)-2sin(x+h/2)sin(h/2))/(h)`


`f'(x)=lim_(h->0)(cos(x+h/2)(sin(h/2))/(h/2))-lim_(h->0)(sin(x+h/2)(sin(h/2))/(h/2))`


`f'(x)=(cos(x+0/2))-(sin(x+0/2))=cosx-sinx`