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

In summary, the problem requires the use of the chain rule and power rule to find the derivative of y = (tan^-1(6x))^2. After applying the power rule, the chain rule is used to find the derivative of tan^-1(6x), which is 1/(1+36x^2). This is then multiplied by the derivative of 2(tan^-1(6x)), giving the final result of 2(tan^-1(6x))*(1/(1+36x^2)).
  • #1
robren
5
0

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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  • #2
I found out how they got 36x^2, but please a detailed review of this problem would still help!
 
  • #3
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.
 
Last edited:

1. What is the Chain Rule and how does it apply to this equation?

The Chain Rule is a calculus principle that allows us to find the derivative of a composite function. In this equation, the function y is composed of two functions: tan^-1(6x) and x^2. The Chain Rule states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

2. How do I find the derivative of the given equation?

To find the derivative of this equation, we can use the Chain Rule. First, we take the derivative of the outer function, which is y = x^2. This gives us 2x. Then, we take the derivative of the inner function, which is tan^-1(6x). This can be rewritten as arctan(6x), and its derivative is 6/(1+(6x)^2). Finally, we multiply these two derivatives to get the final result of 2x * 6/(1+(6x)^2) = 12x/(1+(6x)^2).

3. Can this equation be simplified?

Yes, this equation can be simplified by using the trigonometric identity tan^-1(x) = arctan(x) = atan(x). This allows us to rewrite the equation as y = (atan(6x))^2. Then, we can use the power rule to find the derivative, which gives us 2atan(6x) * 1/(1+(6x)^2) * 6 = 12atan(6x)/(1+(6x)^2).

4. Can this equation be solved for x?

No, this equation cannot be solved for x because it is a composite function. We can find the derivative of this equation, but we cannot find a specific value for x that would make the equation true.

5. How can this equation be applied in real-life situations?

This equation can be applied in real-life situations where a variable is indirectly related to another variable through a function. For example, in physics, the velocity of an object might be represented by a composite function, where the inner function is related to the acceleration and the outer function is related to the time. Understanding the Chain Rule can help us find the derivative of this function and understand the relationship between the variables.

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