frac{d}{dx} { e^x log(1 + x^2) } = | vedclass

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(A) To find the derivative of the product $e^x \log(1 + x^2)$,we use the product rule: $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$.Let $u =

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$e^x \left[ \log(1 + x^2) + \frac{2x}{1 + x^2} \right]$

Vedclass · Answer

(A) To find the derivative of the product $e^x \log(1 + x^2)$,we use the product rule: $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$.Let $u = e^x$ and $v = \log(1 + x^2)$.Then $\frac{du}{dx} = e^x$ and $\frac{dv}{dx} = \frac{1}{1 + x^2} \cdot \frac{d}{dx}(1 + x^2) = \frac{2x}{1 + x^2}$.Applying the product rule:$\frac{d}{dx} \{ e^x \log(1 + x^2) \} = e^x \cdot \frac{d}{dx}(\log(1 + x^2)) + \log(1 + x^2) \cdot \frac{d}{dx}(e^x)$$= e^x \left( \frac{2x}{1 + x^2} \right) + \log(1 + x^2) \cdot e^x$$= e^x \left[ \log(1 + x^2) + \frac{2x}{1 + x^2} \right]$.

$\frac{d}{dx} \{ e^x \log(1 + x^2) \} = $

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