Question

Math

Posted 4 months ago

`What is the derivative of arctan(x)?`

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Answer from Sia

Posted 4 months ago

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 $\frac{1}{1 + x^2}$.

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