given the function f, find the slope of the line tangent to the graph of ( f^{-1} ) at the specified point…

given the function f, find the slope of the line tangent to the graph of ( f^{-1} ) at the specified point on the graph of ( f^{-1} ).\n( f(x)=sqrt{5 x},left(4, \frac{16}{5}\right) )\nthe slope of the tangent line to the graph ( f^{-1} ) is

given the function f, find the slope of the line tangent to the graph of ( f^{-1} ) at the specified point on the graph of ( f^{-1} ).\n( f(x)=sqrt{5 x},left(4, \frac{16}{5}\right) )\nthe slope of the tangent line to the graph ( f^{-1} ) is

Answer

Explanation:

Step1: Find the derivative of (f(x))

Given (f(x)=\sqrt{5x}=(5x)^{\frac{1}{2}}). Using the power rule ((u^n)^\prime = nu^{n - 1}u^\prime), where (u = 5x), (n=\frac{1}{2}), and (u^\prime=5). (f^\prime(x)=\frac{1}{2}(5x)^{-\frac{1}{2}}\times5=\frac{5}{2\sqrt{5x}}).

Step2: Use the formula for the derivative of the inverse function

The formula is ((f^{-1})^\prime(a)=\frac{1}{f^\prime(f^{-1}(a))}). We are given the point ((a,b)=(4,\frac{16}{5})) on the graph of (f^{-1}), so (f^{-1}(4)=\frac{16}{5}). First, find (f^\prime(f^{-1}(4))): substitute (x = \frac{16}{5}) into (f^\prime(x)) (f^\prime(\frac{16}{5})=\frac{5}{2\sqrt{5\times\frac{16}{5}}}=\frac{5}{2\sqrt{16}}=\frac{5}{2\times4}=\frac{5}{8}). Then, ((f^{-1})^\prime(4)=\frac{1}{f^\prime(f^{-1}(4))}).

Answer:

(\frac{8}{5})