# Exploring an Alternative Approach to Implicit Differentiation

• chwala
In summary, the conversation discusses an alternative approach for solving a problem using implicit differentiation. The steps for finding the solution are outlined, and the conversation welcomes any additional insights. The book-solution is presented as a teaching demonstration of using parametric coordinates, but it is noted that sometimes eliminating the parametric coordinates can simplify the problem. An example is provided using the given values, where the derivative is found to be -1/3.
chwala
Gold Member
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.

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.

Last edited:
Or $$\begin{split} \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} &= \begin{pmatrix} t^3 \\ t^2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 3t^2 \\ 2t \\ 0 \end{pmatrix} \times \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \\ &= \begin{pmatrix} t^3 \\ t^2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 2t \\ -3t^2 \\ 0 \end{pmatrix} \end{split}$$ and then $$\lambda = \frac{x - t^3}{2t} = \frac{y - t^2}{-3t^2}\quad\Rightarrow\quad y = t^2 - 3t^2\frac{x - t^3}{2t} = t^2 + \tfrac32 t^4 - \tfrac32 tx.$$

chwala
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.

chwala

## What is implicit differentiation?

Implicit differentiation is a technique used in calculus to find the derivative of a function when the function is not explicitly solved for one variable in terms of another. Instead, the function is given in an implicit form, such as an equation involving both variables. By differentiating both sides of the equation with respect to one variable and applying the chain rule, we can solve for the derivative.

## How does an alternative approach to implicit differentiation differ from the traditional method?

An alternative approach to implicit differentiation may involve different strategies or techniques to simplify the process. For example, it might use a more geometric interpretation, leverage specific algebraic manipulations, or apply numerical methods to approximate derivatives. The goal is to provide a more intuitive understanding or computational efficiency compared to the traditional method.

## Why might someone prefer an alternative approach to implicit differentiation?

Someone might prefer an alternative approach to implicit differentiation if it offers a clearer conceptual understanding, reduces computational complexity, or provides more accurate results in certain contexts. Alternative methods can also be beneficial for handling more complex equations or systems where traditional implicit differentiation becomes cumbersome.

## Can you give an example of an alternative method for implicit differentiation?

One alternative method involves using parametric equations. If a curve is defined parametrically by equations x(t) and y(t), we can find the derivatives dx/dt and dy/dt. Then, the derivative dy/dx can be obtained by dividing dy/dt by dx/dt. This method can be particularly useful when dealing with curves that are difficult to describe explicitly.

## What are the potential drawbacks of using an alternative approach to implicit differentiation?

The potential drawbacks of using an alternative approach to implicit differentiation include increased complexity in understanding the new method, potential limitations in its applicability to certain types of equations, and the need for additional tools or knowledge (e.g., familiarity with parametric equations). Additionally, alternative methods may sometimes provide approximations rather than exact solutions.

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