If y = sin^(-1)(x), what is the second derivative d^2y/dx^2?

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Question: If y = sin^(-1)(x), what is the second derivative d^2y/dx^2?

Options:

Correct Answer: -1/√(1-x^2)^3

Solution:

The second derivative d^2y/dx^2 = -1/√(1-x^2)^3.

If y = sin^(-1)(x), what is the second derivative d^2y/dx^2?

If y = sin^(-1)(x), what is the second derivative d^2y/dx^2?

Step 1: Start with the equation y = sin^(-1)(x). This means y is the inverse sine of x.
Step 2: Find the first derivative dy/dx. The derivative of y = sin^(-1)(x) is dy/dx = 1/√(1-x^2).
Step 3: Now, we need to find the second derivative d^2y/dx^2. To do this, we differentiate dy/dx again.
Step 4: Use the quotient rule or chain rule to differentiate dy/dx = 1/√(1-x^2).
Step 5: The derivative of 1/√(1-x^2) is -1/2 * (1-x^2)^(-3/2) * (-2x) = x/(1-x^2)^(3/2).
Step 6: Therefore, d^2y/dx^2 = -1/√(1-x^2)^3.

Inverse Trigonometric Functions – Understanding the properties and derivatives of inverse trigonometric functions, specifically arcsine in this case.
Chain Rule – Applying the chain rule for differentiation when finding the first and second derivatives.
Higher Order Derivatives – Calculating the second derivative and understanding its implications in the context of the function.