Maxima of Inverse Sine Function?

📌 CONCEPT: The inverse sine function, denoted as sin^(-1) x, is the inverse of the sine function, and its graph shows a maximum value at x = 1/√2.

📐 RULE / FORMULA: To find the derivative of the inverse sine function, we use the formula for the derivative of an inverse function, which states that if y = f^(-1) (x), then f'(y) = 1 / f'(x).

💡 WORKED EXAMPLE: Let's consider the inverse sine function y = sin^(-1) x. We need to find the derivative of this function. Using the formula, we have dy/dx = 1 / (d/dy (sin y)) = 1 / cos y. Now, we need to find the value of y for which dy/dx = 0. Since cos y = 0 at y = π/2, we have dy/dx = 0 at y = π/2. But we want to find the value of x, so we substitute y = π/2 into the equation x = sin y, which gives us x = 1/√2.

⚠️ COMMON MISTAKE: Students may incorrectly assume that the derivative of the inverse sine function is simply the reciprocal of the derivative of the sine function, without considering the chain rule and the formula for the derivative of an inverse function.