Solve Chain Rule Equation: y=(tan^-1(6x))^2

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SUMMARY

The discussion focuses on solving the derivative of the function y = (tan^-1(6x))^2 using the Chain Rule and the Power Rule. The user correctly applies the Power Rule to derive 2(tan^-1(6x)), but struggles with the subsequent application of the Chain Rule, particularly in deriving the denominator. The correct derivative involves recognizing that the derivative of tan^-1(u) requires the substitution of u = 6x, leading to the denominator of 36x^2. A detailed algebraic breakdown is requested to clarify these steps.

PREREQUISITES
  • Understanding of the Chain Rule in calculus
  • Familiarity with the Power Rule for differentiation
  • Knowledge of the derivative of the arctangent function
  • Basic algebraic manipulation skills
NEXT STEPS
  • Review the Chain Rule and its applications in calculus
  • Study the derivative of the arctangent function, specifically d/dx(tan^-1(u))
  • Practice problems involving the Power Rule and Chain Rule together
  • Explore detailed examples of algebraic manipulation in calculus problems
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Students learning calculus, particularly those focusing on differentiation techniques, and anyone seeking clarity on the application of the Chain Rule and Power Rule in complex functions.

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Homework Statement


y = (tan^-1(6x))^2

Homework Equations


Chain Rule, power rule?

The Attempt at a Solution


Okay, so I did power rule to bring it to 2(tan^-1(6x))
Then, I know to use the chain rule...
I get 2(tan^-1(6x)*(1/1+x^2)... I know u = 6x so I play 6 into x^2 and I get 6x^2...
I see everyone else has gotten 36x^2 on the bottom and I have no idea why, can someone please explain this problem IN DETAIL to me because for all the chain rule problems relevant to one like this, it just skips the algebraic steps which I need to know in this...
 
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I found out how they got 36x^2, but please a detailed review of this problem would still help!
 
robren said:

Homework Statement


y = (tan^-1(6x))^2

Homework Equations


Chain Rule, power rule?

The Attempt at a Solution


Okay, so I did power rule to bring it to 2(tan^-1(6x))
Then, I know to use the chain rule...
I get 2(tan^-1(6x)*(1/1+x^2)... I know u = 6x so I play 6 into x^2 and I get 6x^2...
I see everyone else has gotten 36x^2 on the bottom and I have no idea why, can someone please explain this problem IN DETAIL to me because for all the chain rule problems relevant to one like this, it just skips the algebraic steps which I need to know in this...
y' = 2 tan-1(6x) * d/dx(tan-1(6x))

For that last derivative, imagine that it is d/dx(tan-1(u)). What would that be? You need the chain rule for this derivative.
 
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