What is the Derivative of Inverse Secant?

In summary, we are asked to find the derivative of sec^{-1}(\frac{\sqrt{1+x^{2}}}{x}) using the equation sec^{-1}=\frac{U'}{U\sqrt{U^{2}-1}} and the attempt at a solution involved finding U'=-\frac{1}{x^{2}\sqrt{1+x^{2}}} and U\sqrt{U^{2}-1}= \frac{\sqrt{1+x^{2}}}{x^{2}}. The final answer is -\frac{1}{1+x^{2}} and the asker is feeling more confident in their mastery of this skill. The correct answer was not found in their text.
  • #1
biochem850
51
0

Homework Statement



Find the derivative of sec[itex]^{-1}[/itex]([itex]\frac{\sqrt{1+x^{2}}}{x}[/itex])

Homework Equations



sec[itex]^{-1}[/itex]=[itex]\frac{U'}{U\sqrt{U^{2}-1}}[/itex]


The Attempt at a Solution



U'=-[itex]\frac{1}{x^{2}\sqrt{1+x^{2}}}[/itex]



U[itex]\sqrt{U^{2}-1}[/itex]= [itex]\frac{\sqrt{1+x^{2}}}{x^{2}}[/itex]


Therefore the derivative is -[itex]\frac{1}{1+x^{2}}[/itex]


I'm staring to feel more confident with these derivatives and if this is correct, I think I will have come ever closer to mastering this skill. Can someone tell me if this is correct (the answer isn't in my text)?

p.s. I know that the first "U" in the denominator of the relevant equation should be in absolute value brackets but I couldn't find them in the menu.















 
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  • #2
Your answer is right.
 
  • #3
Dick said:
Your answer is right.

Thank you
 

Related to What is the Derivative of Inverse Secant?

1. What is the derivative of inverse secant?

The derivative of inverse secant is given by 1 / (|x| * sqrt(x^2 - 1)).

2. How do you find the derivative of inverse secant?

To find the derivative of inverse secant, you can use the formula 1 / (|x| * sqrt(x^2 - 1)) or use the chain rule with the formula d/dx(sec^-1x) = 1 / (|x| * sqrt(x^2 - 1)).

3. What is the domain of the derivative of inverse secant?

The domain of the derivative of inverse secant is all real numbers except 1 and -1, as these values would result in a division by zero.

4. How do you graph the derivative of inverse secant?

You can graph the derivative of inverse secant by plotting points using the formula 1 / (|x| * sqrt(x^2 - 1)) or using a graphing calculator.

5. What is the significance of the derivative of inverse secant?

The derivative of inverse secant is useful in solving problems related to rates of change, optimization, and finding the maximum or minimum values of a function.

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