Find dy/dx of x^2 − (y^2) x = (3x − 3)y

  • Thread starter Moongn
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In summary, the conversation discussed finding the formula for the slope at every point (x, y) on a graph and the exact slopes of the tangent lines at two points with the x-coordinate of 1. The formula for the slope was determined to be y'(x)= (2x-y^2-3y)/(2xy+3x-3) and it was clarified that the value of x is needed to calculate the slope at a point. The conversation was then moved to the Calculus section.
  • #1
Moongn
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implicitdifferentiation.jpg


x^2−(y^2)x = (3x − 3)y

Using this graph and the equation I need to find two things.

(a) Formula which gives the slope dy/dx at every point (x, y) on the graph.
(b) As you can see in the picture, there are two points on the graph which have x-coordinate equal to 1. What are the exact slopes of the tangent lines at those two points?

I believe I correctly figured out (a) to be y'(x)= (2x-y^2-3y)/(2xy+3x-3)

However, I am not sure how to use this formula to find two different slopes for one value of x.
 
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  • #2
Moongn said:
I believe I correctly figured out (a) to be y'(x)= (2x-y^2-3y)/(2xy+3x-3)
That looks correct

However, I am not sure how to use this formula to find two different slopes for one value of x.
You are given the value of x... what else do you need to calculate the slope at a point?
 
  • #3
Since "find a derivative" is NOT differential equations, I am moving this to the Calculus section.
 
  • #4
I understand now. Thank You!
 

1. What is the purpose of finding dy/dx in this equation?

The purpose of finding dy/dx in this equation is to calculate the rate of change of y with respect to x. This can help us understand the relationship between x and y and how they are affected by each other.

2. How do you find the derivative of x^2 − (y^2) x = (3x − 3)y?

To find the derivative of this equation, we can use the implicit differentiation method. This involves taking the derivative of both sides of the equation with respect to x and solving for dy/dx.

3. What are the steps involved in finding dy/dx?

The steps involved in finding dy/dx are:

  1. Take the derivative of both sides of the equation with respect to x.
  2. Use the product rule, chain rule, and power rule to simplify the derivative.
  3. Isolate dy/dx on one side of the equation.
  4. Substitute in the given values for x and y to find the numerical value of dy/dx.

4. How does the value of dy/dx change as x and y vary in this equation?

The value of dy/dx changes as x and y vary in this equation because the derivative represents the instantaneous rate of change of y with respect to x at a specific point. As the values of x and y change, the slope of the curve (represented by dy/dx) also changes.

5. Can this equation be solved for y in terms of x?

Yes, this equation can be solved for y in terms of x by using algebraic manipulation and solving for y. However, the resulting equation will be in implicit form and cannot be easily graphed or manipulated. It is more useful to find the derivative and use it to understand the relationship between x and y.

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