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 ).

( left.\frac{df^{-1}}{dx}
ight|_{x = 3}=square ) (type an integer or a simplified fraction.)

Explanation:

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

If \(y = f(x)\) has an inverse function \(x = f^{-1}(y)\), then \(\frac{df^{-1}}{dx}\big|_{x = a}=\frac{1}{f^{\prime}(f^{-1}(a))}\).

Step2: Identify the values of \(a\) and \(f^{-1}(a)\)

We are given \(a = 3\). Since the graph of \(y = f(x)\) passes through the point \((2,3)\), then \(f(2)=3\). By the definition of an inverse function, \(f^{-1}(3)=2\).

Step3: Use the formula

We know that \(f^{\prime}(2)=\frac{1}{4}\) (the slope of \(y = f(x)\) at \(x = 2\)). Using the formula \(\frac{df^{-1}}{dx}\big|_{x = 3}=\frac{1}{f^{\prime}(f^{-1}(3))}\), substitute \(f^{-1}(3)=2\) into the formula. So \(\frac{df^{-1}}{dx}\big|_{x = 3}=\frac{1}{f^{\prime}(2)}\).

Step4: Calculate the value

Since \(f^{\prime}(2)=\frac{1}{4}\), then \(\frac{df^{-1}}{dx}\big|_{x = 3}=\frac{1}{\frac{1}{4}} = 4\).

Answer:

\(4\)