Solutions

i) Let $g(x)=x^5$. We know that $g'(x)=5x^4$ by Exercise 4 of this chapter. Hence, we can write

$$f(x)=g(x+3)=(x+3)^5$$

and

$$f'(x)=g'(x+3)=5(x+3)^4$$

by Exercise 8a). This also means that

$$f'(x+3)=5(x+6)^4.$$

ii) Let $z=x+3$. Then $x=z-3$ and

$$f(x+3)=f(z)=(z-3)^5.$$

By the same argument as in i), we have

$$f'(z)=5(z-3)^4$$

and

$$f'(z+3)=5z^4.$$

iii) Let $z=x+3$. Then $x=z-3$ and

$$f(x+3)=f(z)=(z+2)^7.$$

It follows that

$$f'(z)=7(z+2)^6$$

and

$$f'(z+3)=7(z+5)^6.$$