Posted by Ben on Monday, October 27, 2008 at 5:27am.
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^(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)
whoops, I forgot the (1/2). Damon also provided the derivation you wanted.
Related Questions
calculus - Find the derivative of the function using the definition of ...
calculus - You are given the following function. f(x)= (8+x)/(1-8x) (a) Find the...
Calculus - Find the derivative of the function using the definition of a ...
Applied Calculus - Find the first and second derivative of the function: x/(7x+...
calculus - (derivatives) - Find the derivative of the function f by using the ...
Calculus (Derivatives) - What is the derivative of the this functon? g(x) = -500...
Calculus - derivatives - Okay, I want to find the derivative of (x^x)^(x^x)... ...
Calculus - Find the derivative of m(x)=1/(x+1) by using the definition of the ...
Calculus (Derivatives) - Using the product rule, find the derivative of the ...
Calc - I'm creating a "calculu for dummies" project book. I've...
For Further Reading
