How Do You Derive tan^(-1)(sqrt(8x^2-1))?

In summary, to find the derivative of tan^(-1)(sqrt(8x^2-1)), use the chain rule twice. The first chain rule involves the tan^(-1) and the second involves the sqrt. The final answer is 1/(x(sqrt(8x^2-1)). This can be derived by setting f(x) = arctan((sqrt(8x^2-1))) and applying the chain rule to find f'(x).
  • #1
cemar.
41
0
Derivative arctan function! Please help!

Find the derivative of tan^(-1)(sqrt(8x^2-1)))

I knwo that the derivative of tan^(-1)(x) is 1/(1+x^2) and that you are supposed to use the chain rule twice but i cannot seem to get the right answer.
If some one could please show me the steps i could figure out where i went wrong! Thanks!
 
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  • #2
Show us what you did first. The first chain rule involves the tan^(-1) and the second involves the sqrt.
 
  • #3
okay what i did was

1. f(x) = arctan((sqrt(8x^2-1)))

2. f'(x) = ((1/(1+(8x^2-1))) * (1/2)(8x^2-1)^(-1/2) * 16x

3. = 16x/((8x^2) * 2 * sqrt(8x^2-1))

4. = 1/(x(sqrt(8x^2-1))
 
  • #4
I believe that is correct.
 
  • #5
really?! That would explain why i was so confused. These online assignments are no fun. I guess i just have to work on submitting my answers properly. Ill just double check (again).
and thank you so much! I have been tearing my hair out over this for about an hour now trying to figure out where i went wrong.
 

FAQ: How Do You Derive tan^(-1)(sqrt(8x^2-1))?

What is the derivative of the arctan function?

The derivative of the arctan function, also known as the inverse tangent function, is 1/(1+x^2). This can be derived using the quotient rule or by using the fact that arctan(x) is the inverse function of tan(x).

What is the domain and range of the derivative of arctan function?

The domain of the derivative of arctan function is all real numbers, as it is defined for all values of x. The range is also all real numbers, as the derivative can take on any value depending on the input value of x.

How is the derivative of arctan function used in real life?

The derivative of arctan function is commonly used in physics and engineering, particularly in the fields of mechanics and electricity. It is also used in financial mathematics to model growth rates and in signal processing to analyze signals.

What is the relationship between the derivative of arctan function and the derivative of tan function?

The derivative of arctan function is the reciprocal of the derivative of tan function. This means that the derivative of tan function can be found by taking the reciprocal of 1/(1+x^2), which is cos^2(x).

Can the derivative of arctan function be simplified?

Yes, the derivative of arctan function can be simplified using trigonometric identities. For example, 1/(1+x^2) can be rewritten as (1-cos^2(x))/(1+cos^2(x)) and then simplified to -sin(x)/cos^2(x), which is equivalent to -tan(x)/cos(x).

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