How do you find d/dx[f^-1(x)] x=1/2 ?

Question

1261 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-23T17:40:15

# 3/2#.

Suppose that, #f^-1(x)=y :. f(y)=x".............[Defn. of "f^-1]#.

#:." Given that "f(6)=1/2 rArr f^-1(1/2)=6#.

In other words, this means that, when #x=1/2, y=6...(star)#.

#"Now, "(f@f^-1)(x)=f(f^-1(x))=f(y)=x, and #

#(f^-1@f)(y)=f^-1(f(y))=f^-1(x)=y#.

#"The Reqd. Value"=d/dx[f^-1(x)]|_(x=1/2)#,

#=d/dx[y]|_(x=1/2), i.e., #

#=[dy/dx]_(x=1/2)#.

But, it is known that, #dy/dx=1/{dx/dy}#.

#:. "The Reqd. Value"=[dy/dx]_(x=1/2)#,

#=[1/(dx/dy)]_(y=6)............[because, (star)#

#=[1/(d/dy(f(y)))]_(y=6)#,

#=[1/(f'(y))]_(y=6)#,

#=1/(f'(6)#,

#=1/(2/3).........[because," Given]"#.

#rArr "The Reqd. Value"=3/2#.

Enjoy Maths.!