consider the following function. without finding the inverse, evaluate the derivative of the inverse at the…

consider the following function. without finding the inverse, evaluate the derivative of the inverse at the given point.\n$f(x)=ln (2 x+e):(1,0)$\nthe derivative of the inverse at the given point is\n(type an exact answer.)

consider the following function. without finding the inverse, evaluate the derivative of the inverse at the given point.\n$f(x)=ln (2 x+e):(1,0)$\nthe derivative of the inverse at the given point is\n(type an exact answer.)

Answer

Explanation:

Step1: Apply the formula for the derivative of the inverse function

If (y = f(x)) and (x = f^{-1}(y)), then ((f^{-1})^\prime(y)=\frac{1}{f^\prime(x)}), where (y = f(x)). Given the point ((1,0)), when (y = 0), (x = 1) (since if (f(x)=y), for the inverse function (f^{-1}(y)=x)).

Step2: Find the derivative of (f(x))

First, find the derivative of (f(x)=\ln(2x + e)). Using the chain - rule, if (u = 2x+e), then (f(x)=\ln(u)) and (\frac{df}{dx}=\frac{d}{du}(\ln(u))\cdot\frac{du}{dx}). We know that (\frac{d}{du}(\ln(u))=\frac{1}{u}) and (\frac{du}{dx}=2). So (f^\prime(x)=\frac{2}{2x + e}).

Step3: Evaluate (f^\prime(x)) at (x = 1)

Substitute (x = 1) into (f^\prime(x)). Then (f^\prime(1)=\frac{2}{2\times1+e}=\frac{2}{2 + e}).

Step4: Use the formula for ((f^{-1})^\prime(y))

Since ((f^{-1})^\prime(y)=\frac{1}{f^\prime(x)}) and (y = 0,x = 1), ((f^{-1})^\prime(0)=\frac{1}{f^\prime(1)}). Substituting (f^\prime(1)=\frac{2}{2 + e}), we get ((f^{-1})^\prime(0)=\frac{2 + e}{2}).

Answer:

(\frac{e + 2}{2})