Question

method of differentiation question...

Answer 1

let g is the inverse function of f and f'(x) = \[\frac{ x ^{10} }{ 1+x ^{2} }\]
If g(2) = a , then g'(x) is = ?

Answer 2

Okay do you have any ideas on how to solve this?

Answer 3

no..

Answer 4

You can reverse process by integrating f'(x) to get f(x), then find \(f^{-1}(x)\) from f(x)

Answer 5

is there any other method instead of integration becoz i only know some basics of integration...

Answer 6

i'm thinking it is like this..\[g(x)=f'(x)\]

Answer 7

\[g(x)=\frac{ x^{10} }{ 1+x^2 }\]

Answer 8

\[Let~u=x^{10}~and~v=1+x^2\]

Answer 9

\[\frac{ du }{ dx }=10x^9\]\[\frac{ dv }{ dx }=2x\]

Answer 10

in this case,u will hv to use this formula for differentiation\[=\frac{ v\frac{ du }{ dx }-u \frac{ dv }{ dx } }{ v^2 }\]

Answer 11

\[g'(x)=\frac{ (1+x^2)(10x^9)-x^{10}(2x) }{ (1+x^2)^2 }\]

Answer 12

I'm not so sure you can find g'(x) here explicitly. You can find g'(2) for sure using inverse function theorem but not sure about g'(x)

Answer 13

\[g(x) = f^{-1}(x)\]
\[f(g(x)) = f(f^{-1}(x)) = x\]
\[(f(g(x))' = f'(g(x)) \times g'(x) = 1\]
\[g'(x) = \frac{ 1 }{ f'(g(x)) }\]

From this it is possible to work out g'(2) given you know g(2).
However doesn't give you much for all x though unless you know the explicit form of g(x)

Answer 14

so you have

\[g'(x) = \dfrac{1+g^2(x)}{g^{10}(x)}\]

Answer 15

That doesn't look bad at all as you can find \(g'\) whenever you knew \(g\).