suppose that the differentiable function ( y = f(x) ) has an inverse, that the graph of ( f ) passes through…

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

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

Answer

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)