What is a Different Method for Implicit Differentiation?

chwala
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Homework Statement
Find the equation of the normal to a curve given parametric equations;

##x=t^3, y=t^2##
Relevant Equations
Parametric equations
This is a text book example- i noted that we may have a different way of doing it hence my post.

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Alternative approach (using implicit differentiation);

##\dfrac{x}{y}=t##

on substituting on ##y=t^2##

we get,

##y^3-x^2=0##

##3y^2\dfrac{dy}{dx}-2x=0##

##\dfrac{dy}{dx}=\dfrac{2x}{3y^2}##

at points ##(-8,4)##

##\dfrac{dy}{dx}=\dfrac{-1}{3}##

...the rest of the steps to required solution will follow...

...any insight is welcome.
 
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Or \begin{split}<br /> \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} &amp;= \begin{pmatrix} t^3 \\ t^2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 3t^2 \\ 2t \\ 0 \end{pmatrix} \times \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \\<br /> &amp;= \begin{pmatrix} t^3 \\ t^2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 2t \\ -3t^2 \\ 0 \end{pmatrix}<br /> \end{split}<br /> and then <br /> \lambda = \frac{x - t^3}{2t} = \frac{y - t^2}{-3t^2}\quad\Rightarrow\quad<br /> y = t^2 - 3t^2\frac{x - t^3}{2t} = t^2 + \tfrac32 t^4 - \tfrac32 tx.
 
The book-solution is presumably given simply as a teaching-demonstration of how to solve this type of problem using parametric coordinates. But note, sometimes elimination of the parametric coordinates simplifies the problem. In this particular question(at the risk of stating the obvious):

##x=t^3, y= t^2 ⇒ y = x^{\frac 23}##

##\frac {dy}{dx} = \frac 23 x^{-\frac13}##

When ##x = -8, \frac {dy}{dx} = \frac 23 (-8)^{-\frac13}= -\frac 13##

etc.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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