How do you find the derivative of #f(x)=sqrt(x)# ?

The derivative of #sqrt(x)# is #1/(2sqrt(x))#.

Remember that we can rewrite surds like this in index notation. For this case, #sqrtx=x^(1/2)#.

Now we can simply use the power rule for differentiation, namely that #d/dx x^n=nx^(n-1)#. Let #n=1/2#. Therefore:

#d/dx x^(1/2)#
# = 1/2 x^(1/2-1)#

# = 1/2 x^(-1/2)#

#=1/2 xx 1/(sqrtx)#

#=1/(2sqrt(x))#