text: Arctan(y)=3x+y how would I go about finding y'?
  
  • IllmicIll
  In summary, to find y' in the equation arctan(y)=3x+y, you can differentiate it implicitly using the chain rule. The result will be y'/(1+y^2). However, there should not be two y' in the equation, as shown in the example of y=arccot(x^2).
  IllmicIll
  
  Homework Statement
  
  Arctan(y)=3x+y how would I go about finding y'?
  
  The Attempt at a Solution
  
  I tried to start out with...
  
  arctan(y)= tan(x)
  tan(x)=3x+y
  tan(x)-3x=y
  
  y'=1/(1+x^2) -3
  
  is this correct?
  
  IllmicIll said:
  arctan(y)= tan(x)
  
  How you reached here from
  Arctan(y)=3x+y ?
  
  What's
  d/dx (Arctan(y)) ?
  Use the chain rule.
  
  How can arctan(y)=tan(x) when your equation says arctan(y)=3x+y?? That's not right. Don't try to solve for y. You can't. Differentiate it implicitly.
  
  d/dx arctan y
  
  y'/1+y^2*y' ?
  
  IllmicIll said:
  d/dx arctan y
  
  y'/1+y^2*y' ?
  
  Close. But why are there two y' in there? And use parentheses to avoid confusion. There's a difference between 1/1+y^2 and 1/(1+y^2).
  
  y'/(1+y^2) *y'
  I thought the 2nd y' has to be there bc for example
  y=arccot(x^2)
  x^2=cot y
  2x=-csc^2(y) *y'
  
  is that not the case since I already have a y'?
  
  oops nvm...
  
  1. What does the equation Arctan(y)=3x+y mean?
  
  The equation Arctan(y)=3x+y is a mathematical expression that relates the inverse tangent of a variable y to the sum of 3x and y. It is a way to represent a relationship between two variables in a mathematical form.
  
  2. Why is finding y' important in this equation?
  
  In this equation, y' represents the derivative of y with respect to x. Finding y' is important because it allows us to understand the rate of change of y with respect to x. This can help us analyze the behavior of the equation and make predictions about its future values.
  
  3. How can I go about finding y' in this equation?
  
  To find y', we can use the implicit differentiation method. This involves taking the derivative of both sides of the equation with respect to x and using the chain rule to simplify the expression. The resulting expression will give us the value of y' in terms of x and y.
  
  4. Can we use any other method to find y' in this equation?
  
  Yes, there are other methods that can be used to find y' in this equation. For example, we can use the inverse function theorem, which states that the derivative of an inverse function is equal to the reciprocal of the derivative of the original function. We can also use the power rule or the quotient rule, depending on the form of the equation.
  
  5. How does finding y' impact the interpretation of this equation?
  
  Finding y' allows us to interpret the equation in terms of its slope or rate of change. For example, if y' is positive, it means that y is increasing with respect to x. If y' is negative, it means that y is decreasing with respect to x. This information can help us understand the behavior of the equation and make predictions about its values.
  
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explanation: The document is a discussion thread from a homework help forum, specifically focusing on calculus. It contains a math question about finding the derivative \( y' \) in the equation \( \arctan(y) = 3x + y \). The thread includes multiple responses from users, providing guidance and feedback on the user's attempt to solve the problem. The document also includes a series of follow-up questions and explanations to help the user understand the process of implicit differentiation.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Arctan(y)=3x+y how would I go about finding y'?
  
  • IllmicIll
  In summary, to find y' in the equation arctan(y)=3x+y, you can differentiate it implicitly using the chain rule. The result will be y'/(1+y^2). However, there should not be two y' in the equation, as shown in the example of y=arccot(x^2).
  IllmicIll
  
  Homework Statement
  
  Arctan(y)=3x+y how would I go about finding y'?
  
  The Attempt at a Solution
  
  I tried to start out with...
  
  arctan(y)= tan(x)
  tan(x)=3x+y
  tan(x)-3x=y
  
  y'=1/(1+x^2) -3
  
  is this correct?
  
  IllmicIll said:
  arctan(y)= tan(x)
  
  How you reached here from
  Arctan(y)=3x+y ?
  
  What's
  d/dx (Arctan(y)) ?
  Use the chain rule.
  
  How can arctan(y)=tan(x) when your equation says arctan(y)=3x+y?? That's not right. Don't try to solve for y. You can't. Differentiate it implicitly.
  
  d/dx arctan y
  
  y'/1+y^2*y' ?
  
  IllmicIll said:
  d/dx arctan y
  
  y'/1+y^2*y' ?
  
  Close. But why are there two y' in there? And use parentheses to avoid confusion. There's a difference between 1/1+y^2 and 1/(1+y^2).
  
  y'/(1+y^2) *y'
  I thought the 2nd y' has to be there bc for example
  y=arccot(x^2)
  x^2=cot y
  2x=-csc^2(y) *y'
  
  is that not the case since I already have a y'?
  
  oops nvm...
  
  1. What does the equation Arctan(y)=3x+y mean?
  
  The equation Arctan(y)=3x+y is a mathematical expression that relates the inverse tangent of a variable y to the sum of 3x and y. It is a way to represent a relationship between two variables in a mathematical form.
  
  2. Why is finding y' important in this equation?
  
  In this equation, y' represents the derivative of y with respect to x. Finding y' is important because it allows us to understand the rate of change of y with respect to x. This can help us analyze the behavior of the equation and make predictions about its future values.
  
  3. How can I go about finding y' in this equation?
  
  To find y', we can use the implicit differentiation method. This involves taking the derivative of both sides of the equation with respect to x and using the chain rule to simplify the expression. The resulting expression will give us the value of y' in terms of x and y.
  
  4. Can we use any other method to find y' in this equation?
  
  Yes, there are other methods that can be used to find y' in this equation. For example, we can use the inverse function theorem, which states that the derivative of an inverse function is equal to the reciprocal of the derivative of the original function. We can also use the power rule or the quotient rule, depending on the form of the equation.
  
  5. How does finding y' impact the interpretation of this equation?
  
  Finding y' allows us to interpret the equation in terms of its slope or rate of change. For example, if y' is positive, it means that y is increasing with respect to x. If y' is negative, it means that y is decreasing with respect to x. This information can help us understand the behavior of the equation and make predictions about its values.
  
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distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread from a homework help forum, specifically focusing on calculus. It contains a math question about finding the derivative \( y' \) in the equation \( \arctan(y) = 3x + y \). The thread includes multiple responses from users, providing guidance and feedback on the user's attempt to solve the problem. The document also includes a series of follow-up questions and explanations to help the user understand the process of implicit differentiation.
  
  Label: multiple-question-and-answer