Problem:
Find dy/dx, where y = cos^(-1)((x - 1/x)/(x + 1/x))

Solution:

To differentiate y = cos^(-1)((x - 1/x)/(x + 1/x)), we can use the chain rule and the quotient rule.

Step 1: Rewrite the expression
Let's start by simplifying the expression inside the inverse cosine function.

Let A = (x - 1/x)/(x + 1/x)

Step 2: Simplify A
To simplify A, we can multiply both the numerator and the denominator by x:

A = (x^2 - 1)/(x^2 + 1)

Step 3: Differentiate A
Now, let's differentiate A with respect to x using the quotient rule:

dA/dx = [(2x)(x^2 + 1) - (x^2 - 1)(2x)]/(x^2 + 1)^2
= (2x^3 + 2x - 2x^3 + 2x)/(x^2 + 1)^2
= 4x/(x^2 + 1)^2

Step 4: Differentiate y using the chain rule
To differentiate y = cos^(-1)(A), we can use the chain rule:

dy/dx = (dy/du) * (du/dx)

Where u = (x - 1/x)/(x + 1/x)

Step 5: Differentiate cos^(-1)(u)
To differentiate cos^(-1)(u), we can use the formula:

d(cos^(-1)(u))/du = -1 / sqrt(1 - u^2)

Step 6: Substitute u and dA/dx into the chain rule
Now, let's substitute u and dA/dx into the chain rule:

dy/dx = (-1 / sqrt(1 - u^2)) * (4x/(x^2 + 1)^2)

Step 7: Simplify the expression
Finally, let's simplify the expression:

dy/dx = (-4x) / (sqrt(1 - u^2) * (x^2 + 1)^2)

And that's the expression for dy/dx.