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If \( y = \sec^{-1}(x) \), what is \( \frac{dy}{dx} \)?

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Question: If \\( y = \\sec^{-1}(x) \\), what is \\( \\frac{dy}{dx} \\)?

Options:

  1. \\( \\frac{1}{
  2. x
  3. \\sqrt{x^2-1}} \\)
  4. \\( \\frac{1}{x\\sqrt{x^2-1}} \\)
  5. 0
  6. undefined

Correct Answer: x

Solution: 

The derivative of \\( y = \\sec^{-1}(x) \\) is \\( \\frac{1}{|x|\\sqrt{x^2-1}} \\).

If \( y = \sec^{-1}(x) \), what is \( \frac{dy}{dx} \)?

Practice Questions

Q1
If \( y = \sec^{-1}(x) \), what is \( \frac{dy}{dx} \)?
  1. \( \frac{1}{
  2. x
  3. \sqrt{x^2-1}} \)
  4. \( \frac{1}{x\sqrt{x^2-1}} \)

Questions & Step-by-Step Solutions

If \( y = \sec^{-1}(x) \), what is \( \frac{dy}{dx} \)?
  • Inverse Trigonometric Functions – Understanding the derivatives of inverse trigonometric functions, specifically the secant function.
  • Chain Rule – Applying the chain rule in differentiation when dealing with inverse functions.
  • Absolute Value in Derivatives – Recognizing the importance of absolute value in the derivative of the secant inverse function.
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