Question

If $f\left(1\right)=3$ and $f^{\prime }\left(1\right)=-\frac{1}{3}$ then the derivative of ${(x^{11}+f\left(x\right))}^{-2}$ at $x=1$ is

• A. $-\frac{1}{2}$
• B. $-1$
• C. $1$
• D. $f^{\prime }\left(1\right)$

Answer 1

The correct option is D $f^{\prime }\left(1\right)$

The given equation is:

$y=(x^{11}+f(x){)}^{-2}$

Differentiating the equation w.r.t to x once we get,

$\Rightarrow y^{\prime }=-2(x^{11}+f(x){)}^{-3}.(11x^{10}+f^{\prime }\left(x\right))$

Value of the derivative at $x=1$ is given as,

$\Rightarrow y^{\prime }=\frac{-2}{(1+3{)}^3}.(11-\frac{1}{3})$

$\Rightarrow y^{\prime }=\frac{-2}{64}.\frac{32}{3}$

$\Rightarrow y^{\prime }=\frac{-1}{3}$ .....Answer

$\Rightarrow y^{\prime }=f^{\prime }\left(1\right)$