Show that for the curve y=ax^n the ratio
Show that for the curve $y=ax^n$ the ratio of the subtangent to the abscissa of the point of tangency is a constant.
Explanation of Method
Power curves are the archetype of a self-similar tangent: every point sees the same proportions.
Step 1: Differentiate
\[ y = ax^n \;\Longrightarrow\; y' = anx^{n-1}. \tag{1} \]
At $(x_0,y_0)$ with $y_0=ax_0^n$, this gives $y'(x_0)=anx_0^{n-1}$.
Step 2: Compute the subtangent
\[ S_T = \frac{y_0}{y'(x_0)} = \frac{ax_0^n}{anx_0^{n-1}} = \frac{x_0}{n}. \tag{2} \]
Step 3: Form the ratio
\[ \frac{S_T}{x_0} = \frac{x_0/n}{x_0} = \frac{1}{n}. \tag{3} \]
This is independent of $x_0$ — the same value $1/n$ at every point of the curve. $\blacksquare$
Step 4: Verification
\[a=2,\ n=3,\ x_0=1.5\text{: } S_T=x_0/n=0.50000,\ S_T/x_0=0.33333=1/3\ \checkmark\]
\[a=0.5,\ n=-2,\ x_0=2\text{: } S_T/x_0=-0.50000=1/n\ \checkmark\]
Result
Final answer:
\[\boxed{\,\frac{S_T}{x_0} = \frac{1}{n} = \text{const}\,}\]