Price elasticity of demand calculus

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uwulislis

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I need some help for my math investigation on the price elasticity of demand, I am not sure how these two formulas link together, they are both equations for PED, but how does the variable from the first equation changes to the second one?
First equation
{id:8,code:\\begin{align*}\n{\\eta}&={\\left|\\frac{\\frac{dq}{q_{0}}}{\\frac{dp}{p_{0}}}\\right|}\\\\\n{}&={\\frac{dq}{dp}\\frac{p_{0}}{q_{0}}}\t\n\\end{align*},aid:null,backgroundColor:#ffffff,type:align*,font:{color:#000000,size:11,family:Arial},backgroundColorModified:false,ts:1609148203061,cs:4L/yrIlaoK1EaU99ym1wqA==,size:{width:76,height:100}}

Second equation:
1609299648778.png
 
Quantity(q) is a function of price(p), so let q=f(p).

Because of the way calculus was founded (by Newton and Liebniz at about the same time), two different notations are still used today.

If q=f(p), then the first derivative can be symbolised as \(\displaystyle \frac{dq}{dp}\) or as \(\displaystyle f ' (p)\).

So, \(\displaystyle \frac{dq}{dp}\frac{p_0}{q_0} = f ' (p) \frac{p_0}{f(p_0)}\).

They are both saying the same thing, just in a different language.
 
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Note that there is a slight error in both final results. Elasticities are defined as non-negative quantities. The derivative of a normal demand function is non-positive. The absence of absolute value indicators is an error.
 
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