1. The problem is to find the derivative of the inverse function of a given function. 2. The formula for the derivative of the inverse function $f^{-1}$ at a point $y$ is: $$\left(f^{-1}\right)'(y) = \frac{1}{f'\left(f^{-1}(y)\right)}$$ This means the derivative of the inverse at $y$ is the reciprocal of the derivative of the original function evaluated at the inverse function value. 3. Important rules: - The function $f$ must be one-to-one and differentiable. - The derivative $f'(x)$ must not be zero at the point of interest. 4. To find $\left(f^{-1}\right)'(y)$: - First find $x = f^{-1}(y)$, the value such that $f(x) = y$. - Then compute $f'(x)$. - Finally, take the reciprocal $\frac{1}{f'(x)}$. 5. Example: If $f(x) = x^3 + x$, then $f'(x) = 3x^2 + 1$. To find $\left(f^{-1}\right)'(y)$ at some $y$, solve $x^3 + x = y$ for $x$, then compute $\frac{1}{3x^2 + 1}$. This method allows you to find the derivative of the inverse function without explicitly finding the inverse function itself.