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

Answer

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)