Find a formula for $g'''(a)$ Suppose $f$ has third derivation on $\mathbb R$ and for a real number $a$, $f(a)=a$ and $f'(a)=-1$.
Suppose that $g(x)=f(f(x)).$
 Find the value of $g'(a)$ and $g''(a)$.
Find a simple formula for $g'''(a)$.  

Note :  According to the formula for derivation of combination of two functions, I calculated $g'(a)=1$.
We have g''(x)=f''(f(x)).f'(x)^2+f'(f(x)).f''(x)

g''(a)=f''(a)-f''(a) 

g''(a)=0

&amp; similarly calculating g'''(x) , we get

g'''(a)=-2f'''(a)-3f''(a)^2
We have, by the chain rule: $g'(x)=f'(f(x))f'(x)$. Hence

$$ g'(a)=f'(f(a))f'(a)=f'(a)^2=1.$$

Now its your turn to compute $g''(a)$ and $g'''(a)$.