Difficult differentiation question (the concept behind this question is elusive)

[sporus]

Homework Statement

suppose f is a differentiable function such that f(g(x)) = x and f'(x) = 1 + [f(x)]^2. Show that g'(x) = 1/(1+ x^2)

Homework Equations

The Attempt at a Solution

since f(g(x)) = x, i think that f is the inverse of g.
so f = g{inverse}

f'(x) = f'(g(x)) * g'(x) = 1 + [f(x)]^2

we are given g'(x) and f'(x), but i can't make the connection between the fact that f = g{inverse} and how it affects f'(g(x)) because that is the only missing part of the equation.

 

[LoopQG]

f'(x)=1+[f(x)]^2

so f(x)=x+[f(x)^3]/3


so take f'(g(x))

following the rule stated

f'(g(x))=1 + x^2 = 1/g'(x)

notice also f'(g(x))=1 so g(x) must subtract out that x^2

hope this helps

 

[sporus]

we haven't learned any integration yet. we only do derivatives in calc 1 so i might not get marks for solving using integration. is there a way to do this without using integration?

 

[jambaugh]

suppose f is a differentiable function such that f(g(x)) = x and f'(x) = 1 + [f(x)]^2. Show that g'(x) = 1/(1+ x^2)

Here is how I would tackle it work with differentials and use variables instead of functions.

Let y = g(x) so f(y)=x.

g'(x) = dy/dx then f'(y) = dx/dy

Then rewrite: f'(y)=1+f(y)^2 = 1+x^2.
dx/dy = 1+x^2
dy/dx = 1/(1+x^2)

Differentials can be defines so that Leibniz's notation is perfectly consistent even up to taking the reciprocal of a ratio of differentials (which gives the derivative of the inverse function).

 

[fzero]

f'(x)=1+[f(x)]^2

so f(x)=x+[f(x)^3]/3

This is incorrect.

 

[sporus]

Here is how I would tackle it work with differentials and use variables instead of functions.

Let y = g(x) so f(y)=x.

g'(x) = dy/dx then f'(y) = dx/dy

Then rewrite: f'(y)=1+f(y)^2 = 1+x^2.
dx/dy = 1+x^2
dy/dx = 1/(1+x^2)

Differentials can be defines so that Leibniz's notation is perfectly consistent even up to taking the reciprocal of a ratio of differentials (which gives the derivative of the inverse function).

i'll try to wrap my head around this one. thanks
btw, do you think this is a fair question for first year calc?