TITLE: Spivak Calculus Chapter 9 Question 9(ii)
QUESTION [5 upvotes]: I am working out of the 3rd edition. The question is to find $f'(x)$ and $f'(x+3)$ for $f(x+3)=x^5$
My working is the following:
$$\begin{eqnarray}
f(x)&=&f((x+3)-3)\\
&=&(x-3)^5\\
\implies f'(x)&=&5(x-3)^4\\
\end{eqnarray}$$
According to the answers section in the book, this is correct. Now:
$$\begin{eqnarray}
f'(x+3)&=&f'((x)+3)\\
&=&5((x+3)-3)^4\\
&=&5x^4\\
\end{eqnarray}$$
However, the answer in the back of the book for $f'(x+3)$ is $0$. I can't understand how the result $0$ was obtained at all. Is this a problem with the book, or a problem with my working? I can't find an up to date errata list.

REPLY [4 votes]: The answer, of course, is wrong. You must have a very old edition of the book, as I don't remember encountering this in the later editions. Your work is fine. You can also do the second problem by introducing a new function $g$ by setting $f(x+3)=g(x)=x^5$. Then $f'(x+3)=g'(x)=5x^4$. (To see why $f'(x+3)=g'(x)$, just write down the definition of the derivatives.)