Question

If $y=f\left(x\right)=x^3+x^5$ and g is the inverse of f, then find $g^{\prime }\left(2\right)$

• A. $\frac{1}{8}$
• B. $\frac{1}{4}$
• C. $\frac{-1}{8}$
• D. $\frac{-1}{4}$

Answer 1

The correct option is A $\frac{1}{8}$

$\frac{dy}{dx}=3x^2+5x^4$

$\Rightarrow g^{\prime }\left(y\right)=\frac{dx}{dy}=\frac{1}{3x^2+5x^4}$

when $y=2$,

then $2=x^3+x^5\Rightarrow x=1$

$\therefore g^{\prime }\left(2\right)={\frac{dx}{dy}]}_{\underset{y=2}{x=1}}=\frac{1}{3+5}=\frac{1}{8}$

Aliter : $\left(gof\right)\left(x\right)=x$

$g^{\prime }\left[f\right(x\left)\right]f^{\prime }\left(x\right)=1$; when $f\left(x\right)=2$, then $x=1$

$g^{\prime }\left(2\right)\cdot f^{\prime }\left(1\right)=1$ [but $f^{\prime }\left(1\right)=8]$

$\therefore g^{\prime }\left(2\right)=1/8$