Step-by-Step Guide: Finding the Derivative of arccos(x) Using the Chain Rule

d/dx arccos(x)

To find the derivative of arccos(x) with respect to x, we can use the chain rule

To find the derivative of arccos(x) with respect to x, we can use the chain rule. Let’s follow the steps:

1. Start with the function: f(x) = arccos(x)

2. Let y = arccos(x). This implies that x = cos(y), where 0 <= y <= pi. 3. Differentiate both sides of the equation x = cos(y) with respect to x: d/dx (x) = d/dx (cos(y)) 4. The left-hand side is simply 1. 5. The right-hand side requires us to use the chain rule. Recall that the derivative of cos(u) with respect to u is -sin(u): -sin(y) * dy/dx = 1 6. Solve for dy/dx, the derivative of y (arccos(x)) with respect to x: dy/dx = 1/(-sin(y)) 7. Now, we need to express sin(y) in terms of x. Since x = cos(y), we can use the trigonometric identity sin^2(y) = 1 - cos^2(y): sin^2(y) = 1 - x^2 sin(y) = sqrt(1 - x^2) 8. Substitute sin(y) into the derivative dy/dx: dy/dx = 1/(-sqrt(1 - x^2)) Finally, we have found the derivative of arccos(x) with respect to x, which is dy/dx = 1/(-sqrt(1 - x^2)).

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