Derivative of arccos(x): Finding the Rate of Change for the Inverse Cosine Function

d/dx arccos

To find the derivative of the inverse cosine function (arccos), we can start by recalling the definition of the arccos function

To find the derivative of the inverse cosine function (arccos), we can start by recalling the definition of the arccos function.

If y = arccos(x), then cos(y) = x, and we know that the derivative of cos(x) is -sin(x).

To find d/dx arccos(x), we can differentiate both sides of the equation cos(y) = x with respect to x:

-d(sin(y))/dx = dx/dx

Using the chain rule, we can rewrite the left side as:

-d(sin(y))/dy * dy/dx = 1

Since sin(y) is the opposite side divided by the hypotenuse in a right triangle with angle y, we can use the Pythagorean identity sin^2(y) + cos^2(y) = 1 to rewrite -d(sin(y))/dy as -sqrt(1 – cos^2(y)).

Substituting this back into our equation, we have:

-sqrt(1 – cos^2(y)) * dy/dx = 1

Since we are solving for dy/dx, we can rearrange the equation:

dy/dx = -(1 / sqrt(1 – cos^2(y)))

Since cos(y) = x (as defined above), we can substitute cos(y) with x in our equation:

dy/dx = -(1 / sqrt(1 – x^2))

Therefore, the derivative of arccos(x) with respect to x is:

d/dx arccos(x) = -(1 / sqrt(1 – x^2))

This is the general formula for differentiating the inverse cosine function.

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