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The length of the subtangent at any point $\left(x_1, y_1\right)$ on the curve $y=5^x$ is
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Verified Answer
The correct answer is:
$\frac{1}{\log _e 5}$
Given, $y=5^x$
$\begin{aligned}
\Rightarrow & \frac{d y}{d x} & =5^x \log 5 \\
\Rightarrow & \left(\frac{d y}{d x}\right)_{\left(x_1, y_1\right)} & =5^{x_1} \log 5
\end{aligned}$
$\therefore$ Length of subtangent
$=y_1 \frac{d x}{d y}=\frac{y_1}{5^{x_1} \log 5}=\frac{5^{x_1}}{5^{x_1} \log 5}=\frac{1}{\log 5}$
$\begin{aligned}
\Rightarrow & \frac{d y}{d x} & =5^x \log 5 \\
\Rightarrow & \left(\frac{d y}{d x}\right)_{\left(x_1, y_1\right)} & =5^{x_1} \log 5
\end{aligned}$
$\therefore$ Length of subtangent
$=y_1 \frac{d x}{d y}=\frac{y_1}{5^{x_1} \log 5}=\frac{5^{x_1}}{5^{x_1} \log 5}=\frac{1}{\log 5}$
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