Qual é a derivada de #arctan (x / 2) #?

differentiate using the #color(blue)"chain rule"#

#color(red)(|bar(ul(color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|)))........ (A)#

Note that #x/2=1/2x#

let #color(blue)(u=1/2x)rArr(du)/(dx)=1/2#

#color(orange)"Reminder" color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(arctanx)=1/(1+x^2))color(white)(a/a)|)))#

and #y=arctancolor(blue)(u)rArr(dy)/(du)=1/(1+color(blue)(u)^2#

Substitute these values into (A) and convert u back into terms of x.

#dy/dx=1/(1+(x/2)^2)xx1/2=(1/2)/((1+x^2/4))=(1/2)/(1/4(4+x^2))#

#rArrdy/dx=2/(4+x^2)#