What is the derivative of arctan(x)?

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the second derivative of $f(x) = \arctan(x)$, we start by finding the first derivative. The first derivative of $\arctan(x)$ is $\frac{d}{dx}(\arctan(x)) = \frac{1}{1 + x^2}$. ‖ ‖ step 2 ⋮ Now, we need to find the second derivative of $f(x) = \arctan(x)$. We use the quotient rule for differentiation: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$. Here, $u = 1$ and $v = 1 + x^2$. ‖ ‖ step 3 ⋮ Compute the derivatives $u' = 0$ and $v' = 2x$. ‖ ‖ step 4 ⋮ Apply the quotient rule: $\frac{d}{dx}\left(\frac{1}{1 + x^2}\right) = \frac{0 \cdot (1 + x^2) - 1 \cdot 2x}{(1 + x^2)^2} = \frac{-2x}{(1 + x^2)^2}$. ‖ ‖ step 5 ⋮ Therefore, the second derivative of $f(x) = \arctan(x)$ is $\frac{d^2}{dx^2}(\arctan(x)) = \frac{-2x}{(1 + x^2)^2}$. ‖ ∻Answer∻ ⚹ $\frac{-2x}{(1 + x^2)^2}$ ⚹ ∻Key Concept∻ ⚹Second derivative of $\arctan(x)$⚹ ∻Explanation∻ ⚹The second derivative of $\arctan(x)$ is found using the quotient rule on the first derivative $\frac{1}{1 + x^2}$.⚹◊What is the integral of arctan(x)?⍭ Generate me a similar question◊

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