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: Recall 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)).

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

Given (f(x)=\ln(2x + e)). Using the chain - rule ((\ln(u))^\prime=\frac{u^\prime}{u}), where (u = 2x+e) and (u^\prime=2). So (f^\prime(x)=\frac{2}{2x + e}).

Step3: Determine the value of (x) corresponding to the given point ((y,x)=(1,0))

We know that when (y = 1) (the (y) - value of the point ((1,0))), (x = 0) (from the point ((1,0))).

Step4: Evaluate (f^\prime(x)) at (x = 0)

Substitute (x = 0) into (f^\prime(x)): (f^\prime(0)=\frac{2}{2\times0 + e}=\frac{2}{e}).

Step5: Use the formula for ((f^{-1})^\prime(y))

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

Answer:

(\frac{e}{2})