If $f(1)=3, f'(1)=-\frac{1}{3}$, then the derivative of $\left\{x^{11}+f(x)\right\}^{-2}$ at x = 1, is
Answer & explanation
Correct answer: option 4
Let $g(x)=\left\{x^{11}+f(x)\right\}^{-2}$. Then,
$g^{\prime}(x)=-2\left\{x^{11}+f(x)\right\}^{-3}\left\{11 x^{10}+f^{\prime}(x)\right\}$
$\Rightarrow g^{\prime}(1)=-2\{1+f(1)\}^{-3}\left\{11+f^{\prime}(1)\right\}$
$\Rightarrow g^{\prime}(1)=-2(1+3)^{-3}\left(11-\frac{1}{3}\right)=-\frac{1}{3}=f^{\prime}(1)$