How is the derivative of cos inverse(x) expressed?

The derivative of the inverse cosine function, or cos inverse(x), is derived from the relationship between the function and its inverse. Specifically, the derivative of cos inverse(x) with respect to x reflects how changes in x relate to changes in the angle whose cosine is x.

To compute the derivative, we use the identity of the derivatives of inverse trigonometric functions. For cos inverse(x), the derivative is given by:

[

\frac{d}{dx} \text{cos}^{-1}(x) = -\frac{1}{\sqrt{1-x^2}}

]

This expression, -1/sqrt(1 - x^2), indicates that as x approaches the boundaries of its domain (-1 to 1), the rate of change of cos inverse(x) becomes more pronounced. The square root in the denominator indicates that the derivative will be undefined at the endpoints where x equals -1 or 1, reflecting the nature of the cosine function where these angles correspond to critical changes in the behavior of the inverse.

This relationship illustrates that as x moves from -1 to 1, the output of the cos inverse function changes, but the rate of that change is always negative, confirming the negative sign in the derivative.