The length of the subtangent at any point (x_ | vedclass

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(D) Given the curve $y = 5^x$. First,find the derivative $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{d}{dx}(5^x) = 5^x \log_e 5$. At the point $(x_1, y_1)

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$\frac{1}{\log_e 5}$

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(D) Given the curve $y = 5^x$. First,find the derivative $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{d}{dx}(5^x) = 5^x \log_e 5$. At the point $(x_1, y_1)$,the slope of the tangent is $\left(\frac{dy}{dx}\right)_{(x_1, y_1)} = 5^{x_1} \log_e 5$. The length of the subtangent is given by the formula $\left| \frac{y_1}{\frac{dy}{dx}} \right|$. Substituting the values: Length of subtangent $= \frac{y_1}{5^{x_1} \log_e 5}$. Since the point $(x_1, y_1)$ lies on the curve $y = 5^x$,we have $y_1 = 5^{x_1}$. Therefore,the length of the subtangent $= \frac{5^{x_1}}{5^{x_1} \log_e 5} = \frac{1}{\log_e 5}$.

The length of the subtangent at any point $(x_1, y_1)$ on the curve $y=5^x$ is

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