Posted by **Ben** on Monday, October 27, 2008 at 5:27am.

f(a) = a + √a

(a) Find the derivative of the function using the definition of derivative.

- Calculus derivatives -
**drwls**, Monday, October 27, 2008 at 6:23am
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)

- Calculus derivatives -
**Damon**, Monday, October 27, 2008 at 6:42am
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^(n-1) q + n(n-1)/2! p^(n-2)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)

- Calculus derivatives -
**drwls**, Monday, October 27, 2008 at 7:18am
whoops, I forgot the (1/2). Damon also provided the derivation you wanted.

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