Question

In Exercises \(32-34,\) use the inverse function-inverse cofunction identities to derive the formula for the derivative of the function. arccosecant

Short answer

The derivative of the function arccosecant or Csc\(^{-1}\)x is given by \[- \frac{1}{|x| \sqrt{x^2 -1}}\]

Step-by-step solution

Set Up Function and Its Derivative

First, set up the function and its derivative. We denote Csc\(^{-1}\)x or arccsc(x) is the arc-cosecant function, which is an inverse function of the cosecant function. The given task is to find its derivative. Write the formula as follows: \[x = csc(f(x))\]

Implicit Differentiation

Second, apply implicit differentiation. Here the challenge is to differentiate x with respect to f(x). Deriving both sides with respect to f gives: \[-csc(f(x)) cot(f(x)) f'(x) = 1\]

Solve for f'(x)

Third, solve for f'(x). Rearranging the expression and solving it for f’(x) provides a generic formula for the derivative of the arccosecant function. This gives \[f'(x) = - \frac{1}{|x| \sqrt{x^2 -1}}\]

Simplify the Expression

Finally, the answer can be simplified by substituting f(x) back in to give the formula for the derivative of the arccosecant function.