Q107 / 180 · COMEDK 2016 · Mathematics
If $f(x)$ and $g(x)$ are two functions with $g(x)=x-\frac{1}{x}$ and $f o g(x)=x^{3}-\frac{1}{x^{3}}$, then $f^{\prime}(x)=$
- A$3 x^{2}+\frac{3}{x^{4}}$
- B$1+\frac{1}{x^{2}}$
- C$x^{2}-\frac{1}{x^{2}}$
- Correct answerD$3 x^{2}+3$
Solution
$f\{g(x)\}=x^{3}-\frac{1}{x^{3}}$
$\begin{aligned}
f\left(x-\frac{1}{x}\right) &=\left(x-\frac{1}{x}\right)\left(x^{2}+\frac{1}{x^{2}}+1\right) \\
&=\left(x-\frac{1}{x}\right)\left\{\left(x-\frac{1}{x}\right)^{2}+2+1\right\}
\end{aligned}$
$f\left(x-\frac{1}{x}\right)=\left(x-\frac{1}{x}\right)\left\{\left(x-\frac{1}{x}\right)^{2}+3\right\}$
Replace $x-\frac{1}{x}$ by $x$
$f(x)=(x)\left(x^{2}+3\right)=x^{3}+3 x$
$f^{\prime}(x)=3 x^{2}+3$
Hence, the correct answer is option: 'D'.