Two students attempt to prove that dxd[f(x)+g(x)]=f′(x)+g′(x).
Student A writes:
$\displaystyle \frac{f(x+h)+g(x+h)-f(x)-g(x)}{h}
= \frac{f(x+h)-f(x)}{h} + \frac{g(x+h)-g(x)}{h}$,
and then takes the limit h→0.
Student B writes:
$\displaystyle \frac{f(x+h)+g(x)-f(x)-g(x+h)}{h}
= \frac{f(x+h)-f(x)}{h} - \frac{g(x+h)-g(x)}{h}$,
and concludes f′(x)−g′(x).
Which statement about their work is correct?
a) Only Student B’s proof is correct.
b) Both students’ proofs are correct.
c) Only Student A’s proof is correct.
d) Neither student’s proof is correct.