Question

If the function $f\left(x\right)=x^3+e^{x/2}$ and $g\left(x\right)=f^{-1}\left(x\right)$; then the value of g'(1) is

• A. $2$
• B. $-2$
• C. $1$
• D. $0$

Answer 1

The correct option is A $2$

$f\left(x\right)=x^3+e^{x/2}$
$f^{\prime }\left(x\right)=3x^2+\frac{1}{2}{e}^{x/2}$
Given g is inverse of f $\Rightarrow g\left(f\right(x\left)\right)=x$
Differentiating both sides w.r.t $x$
$g^{\prime }\left(f\right(x\left)\right).f^{\prime }\left(x\right)=1\Rightarrow g^{\prime }\left(f\right(x\left)\right)=\frac{1}{f^{\prime }\left(x\right)}$
Clearly $f^{\prime }\left(0\right)=1\therefore g^{\prime }\left(1\right)=g^{\prime }\left(f\right(0\left)\right)=\frac{1}{f^{\prime }\left(0\right)}=2$