Elasticity of a Function as a Exponential Constant I am looking for an explanation as to why the the elasticity of a function of the form $\\f(x)=Cx^\alpha$, where $\\C\gt0$, is a constant $\alpha$.
Furthermore, if anyone also knows of any material that expounds on this, that would also be much appreciated. 
 A: The elacsticity of a differentiable function $y=f(x)$ is
$$\large{\mathcal E_{y,x}}=\frac{dy}{dx}\cdot\frac{x}{y}=f^{'}(x)\cdot\frac{x}{y}$$
If $f(x)=C\cdot x^{\alpha}$ then the derivative w.r.t $x$ is $f^{'}(x)=\alpha\cdot C\cdot x^{\alpha-1}$
Therefore $\large{\mathcal E_{y,x}}=\alpha\cdot C\cdot x^{\alpha-1}\cdot\frac{x}{C\cdot x^{\alpha}}$
It is $x^{\alpha-1}=\frac{x^{\alpha}}{x}$. 
$\large{\mathcal E_{y,x}}=\alpha\cdot C\cdot \frac{x^{\alpha}}{x}\cdot\frac{x}{C\cdot x^{\alpha}}$
Finally $x^{\alpha}, C$ and $x$ can be cancelled. 

The elasticity is defined as the relative change of the dependent variable divided by the relative change of the independent variable.
$\large{\frac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}}\normalsize = \frac{\Delta y}{\Delta x}\cdot \frac{x}{y}$
$\Delta x$ is the distance between two x-values. This is the definition of the elasticity. If $\Delta x \to 0$ then $\frac{\Delta y}{\Delta x}$ becomes $\frac{dy}{dx}$.  We want a constant elasticity $\alpha$. This condition  we can regard and see what kind of function we will get.  The equation is 
$$\frac{dy}{dx}\cdot \frac{x}{y}=\alpha$$
This is a $\texttt{first order linear ordinary differential equation}$ which can be solved. Multiplying both sides by $dx$ and dividing both sides by $x$.
$$\frac1y \, dy=\alpha\cdot \frac1x \,dx $$
Integrating both sides
$\int \frac1y \, dy=\int \alpha\cdot \frac1x \,dx $
$\int \frac1y \, dy=\alpha\cdot\int  \frac1x \,dx $
$\ln(y)=\alpha\cdot ln(x)+c$
Taking both sides as exponents of $e$ 
$$e^{\ln(y)}=e^{\alpha\cdot ln(x)+c}$$
$$y=e^{\alpha\cdot ln(x)}\cdot e^{c}$$
$$y=\left(e^{ ln(x)}\right)^{\alpha}\cdot e^{c}$$
$$y=x^{\alpha}\cdot e^{c}$$
With the substitution $e^c=C$ we get
$$y=C\cdot x^{\alpha}$$
