The second derivative of the function $y = \arctan(x)$ can be computed as follows:

Step 1: First Derivative

The derivative of $\arctan(x)$ is: $\frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2}.$

Step 2: Second Derivative

To find the second derivative, differentiate $\frac{1}{1 + x^2}$ with respect to $x$. Use the quotient rule or rewrite it for clarity: $\frac{1}{1 + x^2} = (1 + x^2)^{-1}.$

Using the chain rule: $\frac{d}{dx} \left( (1 + x^2)^{-1} \right) = -1 \cdot (1 + x^2)^{-2} \cdot \frac{d}{dx}(1 + x^2).$

Now compute $\frac{d}{dx}(1 + x^2) = 2x$, so: $\frac{d}{dx} \frac{1}{1 + x^2} = -\frac{2x}{(1 + x^2)^2}.$

Thus, the second derivative of $\arctan(x)$ is: $\frac{d^2}{dx^2} \arctan(x) = -\frac{2x}{(1 + x^2)^2}.$

Final Answer:

$\boxed{-\frac{2x}{(1 + x^2)^2}}$

Would you like me to explain this process further or show another example?

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Related Questions:

1. What is the third derivative of $\arctan(x)$?
2. How do you find higher-order derivatives of $\arctan(x)$?
3. What is the Taylor series expansion for $\arctan(x)$?
4. How is $\arctan(x)$ used in integrals, such as $\int \frac{1}{1+x^2} \, dx$?
5. How does the graph of $\arctan(x)$ relate to its derivatives?

Tip:

For derivatives involving powers and chain rules, organizing each step helps ensure no details are overlooked!