UPSC NDA Math Model Paper 8

Given: g(x) is the inverse function f(x)
Therefore, g(x)=f−1(x)
⇒f[g(x)]=x
Differenating with respect to x, we get
⇒f′[g(x)]×g′(x)=1
∴g′(x)=

1

f′[g(x)]

Given: f′(x)=

1

1+x4

Replace x by g(x)
f′[g(x)]=

1

1+[g(x)]4

Put in the value of f′[g(x)] in equation (1), we get
Hence, g′(x)=1+[g(x)]4