Calculus derivatives
posted by Ben .
f(a) = a + √a
(a) Find the derivative of the function using the definition of derivative.

The derivative f'(a) is the sum of the derivative of a and the derivative of sqrt a. In this case, a is treated as a variable, not a constant.
The answer is 1 + 1/(sqrt a) 
f(a + h) = a + h + (a+h)^1/2
f(a) = a + a^(1/2)
f(a+h)  f(a) = h + (a+h)^1/2  a^1/2
binomial series for q<p: (p+q)^n= p^n +n p^(n1) q + n(n1)/2! p^(n2)q^2 ....
so
for small h
f(a+h)f(a) = h + a^1/2 + (1/2) a^(1/2)h + (1/2)(1/2)/2*a^(1.5)h^2.. a^1/2)
f(a+h)f(a) = h + (1/2)a^(1/2) h  1/8 a^1.5 h^2 ....
divide by h
(f(a+h)f(a))/h = 1 + (1/2) a^1/2  (1/8) a^1.5) h ...
let h>0
df/da> 1 + (1/2)a^(1/2) 
whoops, I forgot the (1/2). Damon also provided the derivation you wanted.