If y = sin^(-1)(x), then what is the derivative dy/dx?

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

Options:

Correct Answer: 1/√(1-x^2)

Solution:

The derivative of y = sin^(-1)(x) is dy/dx = 1/√(1-x^2).

If y = sin^(-1)(x), then what is the derivative dy/dx?

If y = sin^(-1)(x), then what is the derivative dy/dx?

Step 1: Understand that y = sin^(-1)(x) means y is the angle whose sine is x.
Step 2: Recall the relationship between sine and its inverse. If y = sin^(-1)(x), then x = sin(y).
Step 3: Differentiate both sides of the equation x = sin(y) with respect to x. This gives us 1 = cos(y) * (dy/dx).
Step 4: Solve for dy/dx. Rearranging the equation gives dy/dx = 1/cos(y).
Step 5: Use the Pythagorean identity to express cos(y) in terms of x. Since sin^2(y) + cos^2(y) = 1, we have cos(y) = √(1 - sin^2(y)) = √(1 - x^2).
Step 6: Substitute cos(y) back into the equation for dy/dx. This gives dy/dx = 1/√(1 - x^2).

Inverse Trigonometric Functions – Understanding the derivative of inverse trigonometric functions, specifically arcsine in this case.
Chain Rule – Applying the chain rule when differentiating functions involving inverse trigonometric identities.