Question

If $f(x)$ and $g(x)$ are two functions with $g(x) = x-\frac{1}{x}$ and $fog(x)$ = $x^3-\frac {1}{x^3}$ then $f'(x) =$ is:

• A. $3x^2+\frac {3}{x^4}$
• B. $1+\frac {1}{x^2}$
• C. $x^2+\frac {1}{x^2}$
• D. $3x^2+3$

Answer 1

Answer: (D)

We have, $g(x)=x-\frac{1}{x}$ and $f o g(x)=x^{3}-\frac{1}{x^{3}}$
$\Rightarrow f\{g(x)\}=\left(x-\frac{1}{x}\right)^{3}+3\left(x-\frac{1}{x}\right)$
As $g(x)=x-\frac{1}{x}$ (given)
$\therefore f\left(x-\frac{1}{x}\right)=\left(x-\frac{1}{x}\right)^{3}+3\left(x-\frac{1}{x}\right)$
Let $ x-\frac{1}{x}=t$
$\Rightarrow f(t)=t^{3}+3 t$
$\Rightarrow f(x)=x^{3}+3 x$
$\Rightarrow f'(x)=3 x^{2}+3$