Question

If $ g(x) $ is the inverse of $ f(x) $ and $ f'(x)=\cos x, $ then $ g'(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'(x)=cos\,\,x $ ..(i)
$ \Rightarrow $ $ g(x)={{f}^{-1}}(x) $
$ \Rightarrow $ $ x=f(g(x)) $
On differentiating
$ \Rightarrow $ $ 1=f'\,(g(x)).g'(x) $
$ \Rightarrow $ $ g'(x)=\frac{1}{f'(g(x))}=\frac{1}{\cos \,(g(x))} $
[from Eq.(i)]
$ \Rightarrow $ $ g'(x)=sec\,g\,(x) $