Find an equation of the tangent line to the graph of the function f

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Homework Help Overview

The problem involves finding the equation of the tangent line to the graph of a function defined implicitly by the equation (x - y - 1)³ = x at the point (1, -1). The discussion centers around the application of implicit differentiation and the correct interpretation of the given equation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the use of implicit differentiation to find the derivative and the slope of the tangent line. There are attempts to derive the equation and questions about the relevance of the transformed equation x³ - y³ = 1. Some participants express confusion about the differentiation process and the implications of the equation's structure.

Discussion Status

The discussion is ongoing, with participants exploring different methods of differentiation. Some have suggested returning to notes for clarification, while others are questioning the need for implicit differentiation versus solving for y directly. There is no explicit consensus on the best approach yet.

Contextual Notes

Participants note that they have recently learned about derivatives, which may contribute to their confusion. There is also mention of the challenges posed by the equation's form and the differentiation process.

A_Munk3y
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Homework Statement


Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point.
(x - y - 1)3 = x; (1, -1)

The Attempt at a Solution


x3-y3=1
3y2(dy/dx)-3x2=0
3y2(dy/dx)=3x22
(dy/dx)=3y2/3x2
(dy/dx)=x2/y2
slope = (dy/dx) = 1

y-(-1)= 1(x-1)
y=x-2

thats wrong -.-
i tried making it dy/dx = x/y and still wrong.
where am i messing up? I'm pretty sure it has to do with the x at the end but i have no idea what to do with it!
We just learned derivatives so I'm still messing up with them.
 
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What has x^3-y^3=1 got to do with the problem? Your equation is (x-y-1)^3=x. Find y' using implicit differentiation.
 
Yea, i thought i was deriving it wrong :) Ok, let me go back and look at my notes to see how to do implicit differentiation.
Thanks
 
... i can't figure out how to do the implicit differentiation on this. Could someone help me out with it?
 
A_Munk3y said:
... i can't figure out how to do the implicit differentiation on this. Could someone help me out with it?

The derivative of u^3 is 3*u^2*u'. Put u=x-y-1. What do you get? Is that the part that's confusing you?
 
I'm getting confused on the dx/dy part. I'm not even sure i understand how to do this right.
I thought i had to have them all cubed then get the derivative of x3-y3-13 but obviously that's not right since your first response was what did x3-y3= 1 have to do anything.

So is it 3x2*u and 3y2*u -1? and then i add the dx/dy part somewhere? Or am i just way off here :(
Sorry, we just learned this stuff today and it still hasn't really sunk in.
 
Why would you implicitly differentiate? Since 3 is an odd power, you can solve for y without losing any information...
 
Char. Limit said:
Why would you implicitly differentiate? Since 3 is an odd power, you can solve for y without losing any information...

Good point. If you aren't comfortable with implicit differentiation, try it that way. They both work.
 

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