Question

The length of sub-tangent at $(x_1,y_1)$ to the curve $y=e^{\frac{x}{5}}$ is

Answer 1

Here, $y=e^{\frac{x}{5}}\cdots \left(i\right)$
and $(x_1,y_1)$ lies on the curve.
$\Rightarrow \frac{dy}{dx}=e^{\frac{x}{5}}\cdot \frac{1}{5}\cdots \left(ii\right)$
And we know that the length of the sub-tangent $=\left|y_1\frac{dx}{dy}\right|$
$=|e^{\frac{x_1}{5}}\cdot \frac{5}{e^{\frac{x_1}{5}}}|=5$ units [using $\left(i\right)$ and $\left(ii\right)$]