Assign: y=(1+x^(2))Arctan(x)-x

Identify Components and Rules: Identify the function components and apply the product and chain rules. y = (1 + x^2) * arctan(x)^2 - x dy/dx = d/dx[(1 + x^2) * arctan(x)^2] - d/dx[x]Apply Product and Chain Rules: Differentiate (1 + x^2) * arctan(x)^2 using the product rule. Let u = 1 + x^2 and v = arctan(x)^2. du/dx = 2x, dv/dx = 2 * arctan(x) * (1/(1 + x^2)) dy/dx = u * dv/dx + v * du/dx dy/dx = (1 + x^2) * 2 * arctan(x) * (1/(1 + x^2)) + arctan(x)^2 * 2xDifferentiate Using Product Rule: Simplify the expression. dy/dx = 2 * arctan(x) + 2x * arctan(x)^2 - 1

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Q. Assign:

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y=\left(1+x^{2}\right) \operatorname{Arctan}{x}-x

Identify Components and Rules: Identify the function components and apply the product and chain rules. $\newline$ $y = (1 + x^2) \cdot \text{arctan}(x)^2 - x$ $\newline$ $\frac{dy}{dx} = \frac{d}{dx}[(1 + x^2) \cdot \text{arctan}(x)^2] - \frac{d}{dx}[x]$
Apply Product and Chain Rules: Differentiate $(1 + x^2) \cdot \arctan(x)^2$ using the product rule. $\newline$ Let $u = 1 + x^2$ and $v = \arctan(x)^2$ . $\newline$ $\frac{du}{dx} = 2x$ , $\frac{dv}{dx} = 2 \cdot \arctan(x) \cdot \left(\frac{1}{1 + x^2}\right)$ $\newline$ $\frac{dy}{dx} = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx}$ $\newline$ $\frac{dy}{dx} = (1 + x^2) \cdot 2 \cdot \arctan(x) \cdot \left(\frac{1}{1 + x^2}\right) + \arctan(x)^2 \cdot 2x$
Differentiate Using Product Rule: Simplify the expression. $\newline$ $\frac{dy}{dx} = 2 \cdot \arctan(x) + 2x \cdot \arctan(x)^2 - 1$