According to the limit definition, for #f(x)=sqrt(x)#
#(df_x)/(dx)=lim_(hrarr0) (f(x+h)-f(x))/h#
#color(white)("XXX")=lim_(hrarr0)(sqrt(x+h)-sqrt(x))/h#
#color(white)("XXX")=lim_(hrarr0)((sqrt(x+h)-sqrt(x)))/h * ((sqrt(x+h)+sqrt(x)))/((sqrt(x+h)+sqrt(x)))#
#color(white)("XXX")=lim_(hrarr0)(x+h-x)/(h * (sqrt(x+h)+sqrt(x)))#
#color(white)("XXX")=lim_(hrarr0) cancel(h)/(cancel(h) * (sqrt(x+h)+sqrt(x)))#
#color(white)("XXX")=lim_(hrarr0)1/(sqrt(x+h)+sqrt(x))#
#color(white)("XXX")=1/(2sqrt(x))#
#color(white)("XXX")=1/2 * 1/sqrt(x)#
#color(white)("XXX")=1/2 x^(-1/2)#
At #x=4#
#color(white)("XXX")(df_x)/(dx)(4) = 1/2 xx 4^(-1/2)#
#color(white)("XXXXXXX")=1/2 xx 1/2#
#color(white)("XXXXXXX")=1/4#
