Question

If $g(x)$ is the inverse of $f(x)$ and $f^{\prime }(x)=\cos x,$ then $g^{\prime }(x)$ is equal to

• A. $\sec x$
• B. $\sec (g(x))$
• C. $\cos (g(x))$
• D. $-\sin (g(x))$

Answer 1

Answer: (B)

Given, $g(x)$ is the inverse of $f(x)$ and $f^{\prime }(x)=cosx$ ..(i)
$\Rightarrow$ $g(x)=f^{-1}(x)$
$\Rightarrow$ $x=f(g(x))$
On differentiating
$\Rightarrow$ $1=f^{\prime }(g(x)).g^{\prime }(x)$
$\Rightarrow$ $g^{\prime }(x)=\frac{1}{f^{\prime }(g(x))}=\frac{1}{\cos (g(x))}$
[from Eq.(i)]
$\Rightarrow$ $g^{\prime }(x)=secg(x)$