Find the derivative: frac{d}{dx} left( frac{e | vedclass

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(B) To find the derivative of $\frac{e^x}{1 + x^2}$,we use the quotient rule: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \fr

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

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(B) To find the derivative of $\frac{e^x}{1 + x^2}$,we use the quotient rule: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$.Let $u = e^x$ and $v = 1 + x^2$.Then $\frac{du}{dx} = e^x$ and $\frac{dv}{dx} = 2x$.Applying the quotient rule:$\frac{d}{dx} \left( \frac{e^x}{1 + x^2} \right) = \frac{(1 + x^2)e^x - e^x(2x)}{(1 + x^2)^2}$.Factor out $e^x$ from the numerator:$= \frac{e^x(1 + x^2 - 2x)}{(1 + x^2)^2}$.Recognizing that $1 + x^2 - 2x = (1 - x)^2$:$= \frac{e^x(1 - x)^2}{(1 + x^2)^2}$.

Find the derivative: $\frac{d}{dx} \left( \frac{e^x}{1 + x^2} \right)$

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