Find the derivative of f(x) = arctan((sqrt(1+x^2) - 1)/x) with respect to x.

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Step-by-Step Solution

Step 1: Substitute $x = \tan\theta$

To simplify the expression inside the \arctan function, we use the substitution x = \tan\theta$. This is a common technique when expressions involve \sqrt{1+x^2}$. From this substitution, we can also write \theta = \arctan x$.

Step 2: Simplify the expression inside $\arctan$

Substitute x = \tan\theta into the expression. Using the identity 1+\tan^2\theta = \sec^2\theta$, we simplify the square root term to |\sec\theta|$.

Step 3: Further simplify the expression

Assuming \sec$\theta$ > 0 (which is true for the principal value branch of \arctan x), we can remove the absolute value. Convert \sec$\theta$ and \tan$\theta$ to \cos$\theta$ and \sin$\theta$ respectively, and simplify the complex fraction.

Step 4: Apply half-angle identities

Using the half-angle identities 1 - \cos\theta = 2\sin^2(\theta/2) and \sin\theta = 2\sin(\theta/2)\cos(\theta/2)$, we can simplify the expression to \tan(\theta/2)$.

Step 5: Substitute back into $f(x)$ and differentiate

Now, substitute \tan(\theta/2) back into the original function f(x)$. Since \arctan(\tan y) = y$, we get f(x) = \theta/2$. Replace \theta with \arctan x to express f(x) in terms of x$. Finally, differentiate f(x) with respect to x$. The derivative of \arctan x is \frac{1}{1+x^2}$.