Let h(x)=xg(x), where g(x)= f^-1 (x). Find h'(5) using table below.

Question

x   |  f(x)   |   f'(x)

2   |   4     |   -1

3   |   5     |    2

5   |   1     |    3

 

Let h(x)=xg(x), where g(x)=the inverse of f(x).

Find h'(5)

Answer 1

h(x) = xg(x)

 

h'(x) = xg'(x) + g(x)

 

h'(5) = 5g'(5) + g(5)

 

     Since f(3) = 5, f^-1(5) = g(5) = 3

 

     [f^-1(5)]^' = 1/[f'(3)] = 1/2 = g'(5)

 

So, h'(5) = 5(1/2) + 3 = 5.5

 

Answer 2

f inverse = invf below

h'=g(x)+x*g'(x)=invf(x)+x*invf'(x)=invf(x)+x*(1/f'(invf(x))

invf(5)=3 because f(3)=5

f'(3)=2

h'(5)=3+5*(1/2)=5.5