Question

1. $y = x cdot csc^{-1}x$

for the following exercises, use the given values to find $(f^{-1})(a)$.

1. $f(pi)=0,f(pi)=-1,a = 0$
2. $f(6)=2,f(6)=\frac{1}{3},a = 2$

Explanation:

Step1: Apply the formula for the derivative of an inverse function

The formula for \((f^{-1})'(a)\) is \((f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}\).
Since \(f(6) = 2\), then \(f^{-1}(2)=6\).

Step2: Substitute the values into the formula

We are given \(a = 2\), and we know that \(f^{-1}(2)=6\) and \(f'(6)=\frac{1}{3}\).
Substituting into the formula \((f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}\), we get \((f^{-1})'(2)=\frac{1}{f'(6)}\).

Step3: Calculate the result

Since \(f'(6)=\frac{1}{3}\), then \((f^{-1})'(2)=\frac{1}{\frac{1}{3}}\).
Using the rule \(\frac{1}{\frac{b}{a}}=\frac{a}{b}\) (\(a = 3\), \(b = 1\)), we have \(\frac{1}{\frac{1}{3}}=3\).

Answer:

\(3\)