Question

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

• A. $\frac{1}{13}$
• B. $1$
• C. $\frac{1}{7}$
• D. $\frac{1}{3}$

Answer 1

Answer: (A)

$\frac{1}{13}$

$g\left(x\right)=f^{-1}\left(x\right)⟹f\left(g\right(x\left)\right)=x$

Differentiating on both sides

$f^{\prime }\left(g\right(x\left)\right)g^{\prime }\left(x\right)=1⟹g^{\prime }(-5)=\frac{1}{f^{\prime }\left(g\right(-5\left)\right)}$

$f^{\prime }\left(x\right)=5x^4+6x^2+2$

$g(-5)=f^{-1}(-5)⟹\left(g\right(-5){)}^5+2(g(-5){)}^3+2\left(g\right(-5\left)\right)=-5⟹g(-5)=-1$

$⟹g^{\prime }(-5)=\frac{1}{f^{\prime }(-1)}=\frac{1}{5+6+2}=\frac{1}{13}$