What's the derivative of #arctan (x/2)#?
1 Answer
Jul 30, 2016
Explanation:
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)#
