This is a problem we did previously on a test; my answer was marked correct, but I am having trouble getting my answer in the same form that my teacher gave it.

The question:
Find the derivative of (x/(x+1))^1/2

His answer: ( (x(x+1))^(1/2) ) / (2x(x+1)^2)

My answer: ((x+1)^(1/2)) / 2((x)^(1/2))((x+1)^2)

It looks to me like his answer is just my answer multiplied by x^(1/2) on the top and bottom, but I don't know how he did that. I'd like to know how to get my answer into the same form as his, because he asked us (as extra credit) to do the same problem using another method, and it would be easier for me to compare answers if mine was in the form he provided.

Any help would be welcome.

Never mind, I figured it out-- he rationalized the denominator.

To find the derivative of the function f(x) = (x/(x+1))^(1/2), we can use the chain rule. The chain rule states that if we have a function g(x) raised to some power n, then its derivative is given by:

(d/dx) [g(x)^n] = n * g(x)^(n-1) * g'(x)

Let's break down how to apply the chain rule to our problem:

1. First, rewrite the function in a different form:
f(x) = (x(x+1)^(-1))^(1/2)
= (x * (x+1)^(-1/2))

2. Now, identify the outer function and the inner function:
Outer function: f(x) = g(x)^(1/2)
Inner function: g(x) = x * (x+1)^(-1/2)

3. Determine the derivative of the inner function, g'(x):
g(x) = x * (x+1)^(-1/2)
Applying the product rule and the chain rule, we get:
g'(x) = 1*(x+1)^(-1/2) + x*d/dx[(x+1)^(-1/2)]
= (x+1)^(-1/2) - (x*(1/2)(x+1)^(-3/2))

4. Substitute the inner function and its derivative into the chain rule formula to find the derivative of the outer function:
(d/dx) [g(x)^(1/2)] = (1/2) * g(x)^(-1/2) * g'(x)
= (1/2) * (x * (x+1)^(-1/2))^(-1/2) * [(x+1)^(-1/2) - (x*(1/2)(x+1)^(-3/2))]

5. Simplify the expression:
(d/dx) [g(x)^(1/2)] = (1/2) * (x * (x+1)^(-1/2))^(-1/2) * [(x+1)^(-1/2) - (x*(1/2)(x+1)^(-3/2))]
= (1/2) * (x * (x+1)^(-1/2))^(-1/2) * [(x+1)^(-1/2) - (x/2)(x+1)^(-3/2)]

6. Finally, simplify the expression further to match the form your teacher gave:
(d/dx) [g(x)^(1/2)] = (1/2) * (x * (x+1)^(-1/2))^(-1/2) * [(x+1)^(-1/2) - (x/2)(x+1)^(-3/2)]
= [(x*(x+1))^(-1/2)] / [2 * (x+1) * (x+1)^(-1/2)] * [(x+1) - (x/2)]
= [(x*(x+1))^(-1/2)] / [2*x*(x+1)^2]

So, the derivative of f(x) is (x*(x+1))^(-1/2) / [2*x*(x+1)^2]. This matches the form your teacher provided.