Quotient rule

Solution :

Since

$$g(x)=f^{-1}(x)$$

we use inverse function differentiation formulas.

First derivative : $$g'(y)=\frac1{f'(x)}$$ where $$y=f(x)$$

At $$x=1$$ we have $$y=f(1)=2$$

Therefore,

$$g'(2)=\frac1{f'(1)}$$

$$=\frac12$$

Now second derivative :

$$g''(y)=-\frac{f''(x)}{(f'(x))^3}$$

Hence,

$$g''(2) = -\frac{f''(1)}{(f'(1))^3}$$

$$= -\frac3{2^3}$$

$$= -\frac38$$

Now for third derivative :

$$g'''(y) = \frac{3(f''(x))^2-f'(x)f'''(x)} {(f'(x))^5}$$

Substituting given values :

$$g'''(2) = \frac{3(3)^2-(2)(4)}{(2)^5}$$

$$= \frac{27-8}{32}$$

$$= \frac{19}{32}$$

Final Answer :

$$\frac{19}{32}$$