Do you know differentiation?

Calculus Level pending

Let $f(x) = x+ \sin x$ and $g(x) = f^{-1} (x)$ . Find $g'(2\pi)$ .

\frac12

\frac13

\frac14

1

1 solution

Tanmay Sinha
Jun 3, 2016

let f-1(x)=g(x)

x=f(g(x))

differentiating both sides w.r.t. x

1 = f '(g(x)) * g'(x)

g'(x) = 1/f'(g(x))

Now , let g(2π)= f-1(2π) = p

therefore f(p) = 2π

hence p = 2π as f(x) =x+sinx

therefore g(2π)=2π

g'(x)=1/f'(2π)

f'(2π) = 2

hence 1/f'(2π) = 1/2 .