Question

suppose that the differentiable function ( y = f(x) ) has an inverse, that the graph of ( f ) passes through the point ( (2,3) ), and it has a slope of ( -\frac{1}{4} ) at that point. find the value of ( \frac{df^{-1}}{dx} ) at ( x = 3 ).
( \frac{df^{-1}}{dx}|_{x = 3}=square ) (type an integer or a simplified fraction.)

Explanation:

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

If \(y = f(x)\) and \(f\) is differentiable with an inverse function \(f^{-1}\), then \((f^{-1})^\prime(x)=\frac{1}{f^\prime(f^{-1}(x))}\).

Step2: Identify the values

We want to find \(\frac{df^{-1}}{dx}\big|_{x = 3}\). Given that the graph of \(y = f(x)\) passes through the point \((2,3)\), so \(f(2)=3\), which implies \(f^{-1}(3)=2\). Also, given that \(f^\prime(2)=-\frac{1}{4}\).

Step3: Apply the formula

Substitute \(x = 3\) into the formula \((f^{-1})^\prime(x)=\frac{1}{f^\prime(f^{-1}(x))}\). We get \(\frac{df^{-1}}{dx}\big|_{x = 3}=\frac{1}{f^\prime(f^{-1}(3))}\). Since \(f^{-1}(3)=2\) and \(f^\prime(2)=-\frac{1}{4}\), then \(\frac{df^{-1}}{dx}\big|_{x = 3}=\frac{1}{-\frac{1}{4}}\).

Answer:

\(-4\)