The "first principles" definition of a derivative is:
#f'(x) = lim_(h to 0) (f(x+h)-f(x))/h#
We are given:
#f(x) = x + sqrtx#
To obtain #f(x+h)#, substitute #x+h# for every x in #f(x)#:
#f(x+h) = x+h + sqrt(x+h)#
Substitute #f(x+h)# and #f(x)# into the definition:
#f'(x) = lim_(h to 0) (x+h + sqrt(x+h)-(x + sqrtx))/h#
Distribute the negative sign:
#f'(x) = lim_(h to 0) (x+h + sqrt(x+h)-x - sqrtx)/h#
Combine like terms:
#f'(x) = lim_(h to 0) (h + sqrt(x+h) - sqrtx)/h#
Separate into two fractions:
#f'(x) = lim_(h to 0) h/h + (sqrt(x+h) - sqrtx)/h#
The first fraction, #h/h#, becomes 1:
#f'(x) = lim_(h to 0) 1 + (sqrt(x+h) - sqrtx)/h#
We can use the property #(a-b)(a+b) = (a^2-b^2)# to eliminate the radicals in the numerator; this means that we must multiply by 1 in the from of #(sqrt(x+h) + sqrtx)/(sqrt(x+h) + sqrtx)#:
#f'(x) = lim_(h to 0) 1 + (sqrt(x+h) - sqrtx)/h(sqrt(x+h) + sqrtx)/(sqrt(x+h) + sqrtx)#
Please observe that we have effectively multiplied by 1 and the radicals in the numerator are eliminated by squaring:
#f'(x) = lim_(h to 0) 1 + (x+h - x)/(h(sqrt(x+h) + sqrtx))#
The #x - x# in the numerator becomes 0 :
#f'(x) = lim_(h to 0) 1 + h/(h(sqrt(x+h) + sqrtx))#
The #h/h# becomes 1:
#f'(x) = lim_(h to 0) 1 + 1/(sqrt(x+h) + sqrtx)#
Now, we may let #h to 0#:
#f'(x) = 1 + 1/(sqrtx + sqrtx)#
Combine like terms:
#f'(x) = 1 + 1/(2sqrtx)#
