The differential of {e^{x^3}} with respect to | vedclass

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(C) Let $u = e^{x^3}$ and $v = \log_e x$. We need to find $\frac{du}{dv} = \frac{du/dx}{dv/dx}$. First,differentiate $u$ with respect to $x$: $\frac{d

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$3x^3 e^{x^3}$

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(C) Let $u = e^{x^3}$ and $v = \log_e x$. We need to find $\frac{du}{dv} = \frac{du/dx}{dv/dx}$. First,differentiate $u$ with respect to $x$: $\frac{du}{dx} = \frac{d}{dx}(e^{x^3}) = e^{x^3} \cdot \frac{d}{dx}(x^3) = e^{x^3} \cdot 3x^2$. Next,differentiate $v$ with respect to $x$: $\frac{dv}{dx} = \frac{d}{dx}(\log_e x) = \frac{1}{x}$. Now,calculate the ratio: $\frac{du}{dv} = \frac{3x^2 e^{x^3}}{1/x} = 3x^2 e^{x^3} \cdot x = 3x^3 e^{x^3}$. Thus,the correct option is $C$.

The differential of ${e^{x^3}}$ with respect to $\log_e x$ is

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