Let $f(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}$, and $g(x)=f^{-1}(x)$. Compute $g^{(3)}(0)$ (the 3rd derivative of g).   I have a solution but its not giving the correct answer. My question is not how to solve the problem but what I did wrong in my solution.  First $g(0)=f^{-1}(0)=0$  $g^{'}(x)= (f^{'}(f^{-1}(x)))^{-1}$, so $g^{'}(0)=1$  $g^{(2)}(x)=-(f^{'}(f^{-1}(x)))^{-2} \cdot f^{(2)}(f^{-1}(x)) \cdot g^{'}(x)= -(g^{'}(x))^{-3} \cdot f^{(2)}(f^{-1}(x))$, so $g^{(2)}(0)=-1$  Finally $g^{(3)}(x)=3(g^{'}(x))^{-4} \cdot g^{(2)}(x) \cdot f^{(2)}(f^{-1}(x))+ f^{(3)}(f^{-1}(x)) \cdot g^{'}(x) \cdot -(g^{'}(x))^{-3}$, so $g^{(3)}(0)=-3-2=-5$. The correct answer is 1, however. I've checked my work so many times but still can't figure out what I did incorrect.
Is my way of solving and my answer correct?  Let $f(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}$   And $g(x)=f^{-1}(x)$  Find $g'''(0)$  My attempt:  We know that $g'(x)=\frac{1}{f'(g(x))}$  $f'(x)=1+x+x^2+x^3+x^4=\frac{x^5-1}{x-1}$  $f'(g(x))=\frac{(g(x))^5-1}{(g(x))-1}$  $\Rightarrow g'(x)=\frac{(g(x))-1}{(g(x))^5-1}$  $\Rightarrow g'(x)[(g(x))^5-1]=g(x)-1$  Similarly differentiating again,  $\Rightarrow g'(x)[5(g(x))^4]+[(g(x))^5-1]g''(x)=g'(x)$  Similarly differentiating again,  $\Rightarrow g'(x)[20(g(x))^3g'(x)]+[5(g(x))^4]g''(x)+[(g(x))^5-1]g'''(x)+g''(x)[5(g(x))^4]=g''(x)$  Putting $x=0$,  $g(0)=f^{-1}(0)=0$  $g'(0)=\frac{1}{f'(g(0))}=\frac{1}{f'(0)}=1$  similarly,$g''(0)=-1$  $\Rightarrow g'''(0)=1$
Since multiple inheritance is bad (it makes the source more complicated) C# does not provide such a pattern directly. But sometimes it would be helpful to have this ability. For instance I'm able to implement the missing multiple inheritance pattern using interfaces and three classes like that: public interface IFirst { void FirstMethod(); } public interface ISecond { void SecondMethod(); }  public class First:IFirst  {      public void FirstMethod() { Console.WriteLine(&quot;First&quot;); }  }  public class Second:ISecond  {      public void SecondMethod() { Console.WriteLine(&quot;Second&quot;); }  }  public class FirstAndSecond: IFirst, ISecond {     First first = new First();     Second second = new Second();     public void FirstMethod() { first.FirstMethod(); }     public void SecondMethod() { second.SecondMethod(); } }  Every time I add a method to one of the interfaces I need to change the class FirstAndSecond as well. 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