Question

What is the derivative of `sin (x/2)`?

Answer 1

`1/2cos(x/2)`

Explanation:

Differentiate using the `color(blue)"chain rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(f(g(x)))=f'(g(x))g'(x))color(white)(a/a)|))) ........ (A)`

`f(g(x))=sin(x/2)rArrf'(g(x))=cos(x/2)`

and `g(x)=x/2=1/2xrArrg'(x)=1/2`
`"------------------------------------------------------"`
Substitute rhese values into (A)

`rArrf'(g(x))=cos(x/2)xx1/2=1/2cos(x/2)`