Question

If is a strictly monotonic differentiable function with $f^1\left(x\right)=\frac{1}{\sqrt{1+x^3}}$ . If g is the inverse of f then

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$y=f\left(x\right)\iff x=f^{-1}\left(y\right)\iff x=g\left(y\right)sinceg=f^{-1}\frac{dx}{dy}=g^1\left(y\right)=\frac{1}{f^1\left(x\right)}=\sqrt{1+x^3}\frac{d^{2}x}{d{y}^2}=g^{11}\left(y\right)={\frac{d}{dy}}^{\frac{dx}{dy}}={\frac{d}{dy}}^{\sqrt{1+x^3}}={\frac{d}{dx}}^{\left(\sqrt{1+x^3}\right)}\times \frac{dx}{dy}=\frac{3x^2}{2\sqrt{1+x^3}}\times g^1\left(y\right)=\frac{3x^2}{2}{g}^{11}\left(y\right)=\frac{3}{2}{g}^2\left(y\right)$