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How do you calculate the derivative of cos inverse x?

When differentiating cos inverse x, the typical method is to make  y equal to cos inverse x.

By taking cos of both sides: x = cosy.

You can then differentiate with respect to y, obtaining that: (dx/dy) = - siny

Using our knowledge of derivatives, we now know that:   (dy/dx) = -1/(siny)

From x = cosy, x^2 = (cosy)^2

                                   = 1 - (siny)^2

                   (siny)^2 = 1 - x^2

                        siny   = (1-x^2)^(1/2)

Combining this with the equation stating (dy/dx), we get:

          (dy/dx) = (-1)/((1-x^2)^(1/2))

Since y is equal to the cos inverse function, this is now equal to the derivative of cos inverse x.

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