Question

1. let ( f ) be differentiable function and ( f(3)=4, f^{prime}(3)=5, f^{prime}(4)=6 ). what is ( left(f^{-1}

ight)^{prime}(4) )? your answer:

Explanation:

Step1: Recall the formula for the derivative of an 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: Identify the value of \(x\) for the given \(y\)

We are given \(y = 4\) and \(f(3)=4\). So when \(y = 4\), \(x = 3\).

Step3: Apply the formula

Using the formula \((f^{-1})^\prime(y)=\frac{1}{f^\prime(x)}\), substitute \(y = 4\) (so \(x = 3\)) and \(f^\prime(3)=5\). Then \((f^{-1})^\prime(4)=\frac{1}{f^\prime(3)}\).

Answer:

\(\frac{1}{5}\)